Startseite New properties for the Ramanujan R-function
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New properties for the Ramanujan R-function

  • Chuan-Yu Cai , Lu Chen , Ti-Ren Huang EMAIL logo und Yuming Chu
Veröffentlicht/Copyright: 1. September 2022
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 20 Heft 1

Abstract

In the article, we establish some monotonicity and convexity (concavity) properties for certain combinations of polynomials and the Ramanujan R-function by use of the monotone form of L’Hôpital’s rule and present serval new asymptotically sharp bounds for the Ramanujan R-function that improve some previously known results.

Keywords: Ramanujan R-function; Gaussian hypergeometric function; Riemann zeta function; generalized elliptic integral; monotonicity; convexity
MSC 2010: 33C05; 33B15; 26A51; 26D20

1 Introduction

Let Re ( a ) > 0 and Re ( b ) > 0 . Then, the classical gamma function Γ ( ⋅ ) [1,2, 3,4], psi function ψ ( ⋅ ) [5,6, 7,8], beta function B ( ⋅ , ⋅ ) [9,10], and Euler-Mascheroni constant γ [11,12,13, 14,15] are defined by

(1.1) Γ ( a ) = ∫ 0 ∞ t a − 1 e − t d t , ψ ( a ) = Γ ′ ( a ) Γ ( a ) , B ( a , b ) = Γ ( a ) Γ ( b ) Γ ( a + b )

and

γ = lim n → ∞ ∑ k = 1 n 1 k − log n = 0.57721566 … ,

respectively, and the Ramanujan R -function R ( ⋅ ) [16,17,18] is defined by

(1.2) R ( a ) = − 2 γ − ψ ( a ) − ψ ( 1 − a ) , a ∈ ( 0 , 1 ) ,

and it is the special case of the two parameter function

(1.3) R ( a , b ) = − 2 γ − ψ ( a ) − ψ ( b ) ,

and sometimes R ( a , b ) is also said to be the Ramanujan constant although it is a function of a and b .

For a ∈ ( 0 , 1 ) , we denote

(1.4) B ( a ) = B ( a , 1 − a ) = Γ ( a ) Γ ( 1 − a ) = π sin ( π a ) .

By the symmetry, in what follows, we assume that a ∈ ( 0 , 1 / 2 ] in (1.2) and (1.4).

For a , b , c ∈ R with c ≠ 0 , − 1 , − 2 , … , the Gaussian hypergeometric function [19,20,21, 22,23,24, 25,26,27, 28,29,30] is defined by

(1.5) F ( a , b ; c ; x ) = F 1 2 ( a , b ; c ; x ) = ∑ n = 0 ∞ ( a ) n ( b ) n ( c ) n x n n ! , x ∈ ( − 1 , 1 ) ,

where ( x ) n denotes the shifted factorial function defined by ( x ) n = x ( x + 1 ) ⋯ ( x + n − 1 ) for n ≥ 1 and ( x ) 0 = 1 for x ≠ 0 . It is well known that F ( a , b ; c ; x ) has wide applications in many branches of mathematics and physics. Many elementary and special functions in mathematical physics are the particular or limiting cases of the Gaussian hypergeometric function. F ( a , b ; c ; x ) is said to be zero-balanced if c = a + b . The asymptotic properties of F ( a , b ; a + b ; x ) are related to B ( a , b ) and R ( a , b ) as x → 1 [31,32,33, 34,35,36]. For example, F ( a , b ; a + b ; x ) satisfies the Ramanujan asymptotic identity [37]:

(1.6) B ( a , b ) F ( a , b ; a + b ; x ) + log ( 1 − x ) = R ( a , b ) + O ( ( 1 − x ) log ( 1 − x ) )

when x → 1 .

Let a , r ∈ ( 0 , 1 ) . Then, the generalized elliptic integral K a ( r ) [38,39,40, 41,42,43, 44,45,46, 47,48,49, 50,51,52] of the first kind is defined by

(1.7) K a ( r ) = π 2 F ( a , 1 − a ; 1 ; r 2 ) , K a ( 0 + ) = π 2 , K a ( 1 − ) = ∞ .

For the sake of simplicity and convenience, in what follows, we denote r ′ = 1 − r 2 for r ∈ [ 0 , 1 ] and K ′ ( r ) = K a ( r ′ ) . By the symmetry, we only need to discuss the case of a ∈ ( 0 , 1 / 2 ] . From (1.6), we obtain the asymptotic identity:

(1.8) B ( a ) K a ( r ) − π log e R ( a ) / 2 r ′ = O ( ( r ′ ) 2 log ( r ′ ) )

when r → 1 .

Recently, the properties and bounds involving K a ( r ) and R ( a ) have attracted the attention of many researchers. Wang et al. [53] proved that the double inequality

(1.9) π R ( a ) + [ B ( a ) − R ( a ) ] r 2 < K a ( r ) log ( e R ( a ) / 2 / r ′ ) < π B ( a ) [ 1 − a ( 1 − a ) + a ( 1 − a ) r 2 ]

holds for all a ∈ ( 0 , 1 / 2 ] and r ∈ ( 0 , 1 ) .

In [54], the authors established the following two-sided inequality:

sin ( π a ) 1 − b + b r 2 log e R ( a ) / 2 r ′ − C ≤ K a ( r ) ≤ sin ( π a ) 1 − b + b r 2 log e R ( a ) / 2 r ′ − C r ′ 2 ,

for all a ∈ ( 0 , 1 / 2 ] and r ∈ ( 0 , 1 ) , where C = [ R ( a ) − ( 1 − b ) B ( a ) ] / 2 and b = a ( 1 − a ) .

Qiu et al. [17] presented the power series expansions of the function R ( a ) at a = 0 and a = 1 / 2 , and provided the monotonicity and convexity properties of certain combinations defined in terms of polynomials and the difference between the Ramanujan R -function R ( a ) and the beta function B ( a ) , and proved that

(1.10) 2 ( 2 a 2 n + α 2 n − 1 ) x 2 n + 1 ≤ B ( x ) − R ( x ) + 2 ∑ k = 1 2 n a k x k ≤ ( 2 a 2 n + α 2 n − 1 ) x 2 n

and

(1.11) ( 2 a 2 n + 1 + α 2 n ) x 2 n + 1 ≤ B ( x ) − R ( x ) + 2 ∑ k = 1 2 n + 1 a k x k ≤ 2 ( 2 a 2 n + 1 + α 2 n ) x 2 ( n + 1 )

for n ∈ N and x ∈ ( 0 , 1 / 2 ] , where

a n = { ( − 1 ) n + [ 1 + ( − 1 ) n + 1 ] 2 − n − 1 } ζ ( n + 1 ) ,

α n = 2 n + 1 π − log 16 + 2 ∑ k = 1 n 2 − k a k

and

(1.12) ζ ( s ) = ∑ k = 1 ∞ 1 k s ( Re ( s ) > 1 )

is the Riemann zeta function [55,56].

Huang et al. [54] established several monotonicity, concavity, and convexity properties for certain combinations defined in terms of R ( a ) and b = a ( 1 − a ) , and proved that

(1.13) ( 7 log 2 − 3 ) b ≤ b ( 2 − b ) R ( a ) + b − 1 ,

(1.14) A 2 b ≤ 1 − b 2 − b ( 1 − b ) R ( a ) < 2 b ,

(1.15) log 2 + D 1 ( 1 − 2 a ) 2 − D 2 ( 1 − 2 a ) 4 − D 3 ( 1 − 2 a ) 5 ≤ b R ( a ) ≤ log 2 + ( 1 − log 2 ) ( 1 − 2 a ) ,

where D 1 = λ ( 3 ) − log 2 , D 2 = λ ( 3 ) − λ ( 5 ) , D 3 = λ ( 5 ) − 1 , A 2 = 15 / 4 − log 8 and

(1.16) λ ( n + 1 ) = ∑ k = 0 ∞ 1 ( 2 k + 1 ) n + 1 .

From [31], one has

(1.17) λ ( n + 1 ) = ( 1 − 2 − n − 1 ) ζ ( n + 1 ) .

We also denote

(1.18) β ( n ) = ∑ k = 0 ∞ ( − 1 ) k 1 ( 2 k + 1 )

in the full article.

The Ramanujan R -function often appears in the studies of the properties of the generalized Grötzsch ring function μ a ( r ) [57,58], the generalized Hersch-Pfluger distortion function φ K a ( r ) [59,60,61], and the generalized η K -distortion function η K a ( t ) [62]. Therefore, the function R ( a ) is indispensable for studying the properties of K a ( r ) and many other special functions [63].

The main purpose of this article is to establish new asymptotically sharp lower and upper bounds for the Ramanujan R -function R ( a ) and to improve inequalities (1.14) and (1.15).

2 Lemmas

To prove our main results, we need several technical lemmas, which we present in this section.

Lemma 2.1

(See [54, Lemma 2.1]) The following two statements are true:

  1. The function λ ( n ) defined by (1.17) is strictly decreasing with respect to n ∈ N \ { 1 } with λ ( 2 ) = π 2 / 8 and lim n → ∞ λ ( n ) = 1 , and the function β ( n ) defined by (1.18) is strictly increasing with respect to n ∈ N with β ( 1 ) = π / 4 and lim n → ∞ β ( n ) = 1 .

  2. The following identities

    (2.1) R ( a ) = 1 a + 2 ∑ n = 1 ∞ ζ ( 2 n + 1 ) a 2 n = log 16 + 4 ∑ n = 1 ∞ λ ( 2 n + 1 ) ( 1 − 2 a ) 2 n = log 16 + 4 ∑ n = 1 ∞ ( 1 − 2 − 2 n − 1 ) ζ ( 2 n + 1 ) ( 1 − 2 a ) 2 n = 1 b − 4 ( 1 − log 2 ) + 4 ∑ n = 1 ∞ [ λ ( 2 n + 1 ) − 1 ] ( 1 − 2 a ) 2 n

    and

    (2.2) B ( a ) = 1 a + 2 ∑ n = 1 ∞ ( 1 − 2 1 − 2 n ) ζ ( 2 n ) a 2 n − 1 = 4 ∑ n = 1 ∞ β ( 2 n + 1 ) ( 1 − 2 a ) 2 n

    hold for all a ∈ ( 0 , 1 / 2 ] .

Lemma 2.2

The function f ( s ) = − 2 ζ ( 2 s + 3 ) + ζ ( 2 s + 5 ) + 9 ζ ( 2 s + 7 ) / 8 is strictly increasing from [ 1 , ∞ ) onto [ f ( 1 ) , 1 / 8 ) , where f ( 1 ) = − 2 ζ ( 5 ) + ζ ( 7 ) + 9 ζ ( 9 ) / 8 = 0.0617 … .

Proof

It follows from (1.12) that

f ( s ) = − 2 ζ ( 2 s + 3 ) + ζ ( 2 s + 5 ) + 9 8 ζ ( 2 s + 7 ) = ∑ k = 1 ∞ − 2 k − ( 2 s + 3 ) + k − ( 2 s + 5 ) + 9 8 k − ( 2 s + 7 ) = ∑ k = 1 ∞ − 2 k − 3 + k − 5 + 9 8 k − 7 k − 2 s = ∑ k = 1 ∞ − 16 k 4 + 8 k 2 + 9 8 k 7 k − 2 s .

Differentiating f ( s ) leads to

f ′ ( s ) = ∑ k = 1 ∞ ( − 2 log k ) − 16 k 4 + 8 k 2 + 9 8 k 7 k − 2 s = ∑ k = 2 ∞ 2 log k 16 k 4 − 8 k 2 − 9 8 k 7 k − 2 s .

Therefore, f ′ ( s ) > 0 for all s > 0 and f is strictly increasing due to 16 k 4 − 8 k 2 − 9 > 0 and log k > 0 for k ≥ 2 . Note that f ( 1 ) = − 2 ζ ( 5 ) + ζ ( 7 ) + 9 ζ ( 9 ) / 8 = 0.06175 … and f ( ∞ ) = 1 / 8 .□

The following monotone form of L’Hôpita’s rule can be found in the literature [[32], Theorem 1.25].

Lemma 2.3

Let − ∞ < a < b < ∞ , f , g : [ a , b ] → R be continuous on [ a , b ] and be differentiable on ( a , b ) , and g ′ ( x ) ≠ 0 on ( a , b ) . If f ′ ( x ) / g ′ ( x ) is increasing (decreasing) on ( a , b ) , then so are the functions

[ f ( x ) − f ( a ) ] / [ g ( x ) − g ( a ) ] , [ f ( x ) − f ( b ) ] / [ g ( x ) − g ( b ) ] .

If f ′ ( x ) / g ′ ( x ) is strictly monotone, then the monotonicity in the conclusion is also strict.

3 Main results

Throughout this paper, we always assume that a ∈ ( 0 , 1 / 2 ] and b = a ( 1 − a ) . Let

(3.1) A 1 ( 2 ) = log 2 − λ ( 3 ) , A 1 ( 2 k ) = λ ( 2 k − 1 ) − λ ( 2 k + 1 ) ( k ≥ 2 ) , B 1 ( 1 ) = 1 , B 1 ( 2 ) = 0 , B 1 ( 2 k − 1 ) = − 2 ζ ( 2 k − 1 ) , B 1 ( 2 k ) = 2 ζ ( 2 k − 1 ) ( k ≥ 2 ) .

Theorem 3.1

Let n ∈ N and f 1 ( a ) = − b R ( a ) = − a ( 1 − a ) R ( a ) . Then one has

  1. f 1 has the power series formula

    (3.2) f 1 ( a ) = − log 2 + ∑ n = 1 ∞ A 1 ( 2 n ) ( 1 − 2 a ) 2 n = − 1 + ∑ n = 1 ∞ B 1 ( n ) a n .

  2. f 1 is increasing, concave, and ( − 1 ) k f 1 ( k ) ( a ) > 0 on ( 0 , 1 / 2 ) for k ≥ 3 .

  3. Let

    (3.3) F 1 , n ( a ) = f 1 ( a ) − P 1 , n ( a ) ( 1 − 2 a ) 2 ( n + 1 ) ,

    where P 1 , n ( a ) = − log 2 + ∑ k = 1 n A 1 ( 2 k ) ( 1 − 2 a ) 2 k and A 1 ( 2 k ) ( k ≥ 1 ) is defined by (3.1). Then, F 1 , n is strictly decreasing and convex from ( 0 , 1 / 2 ) onto ( A 1 ( 2 n + 2 ) , λ ( 2 n + 1 ) − 1 ) . In particular, the double inequality

    (3.4) 0 ≤ − b R ( a ) − P 1 , n ( a ) − A 1 ( 2 n + 2 ) ( 1 − 2 a ) 2 n + 2 ≤ [ λ ( 2 n + 1 ) − 1 − A 1 ( 2 n + 2 ) ] ( 1 − 2 a ) 2 n + 3

    holds for all n ∈ N and a ∈ ( 0 , 1 / 2 ] , with equality in each instance if and only if a = 1 / 2 .

  4. Let

    (3.5) G 1 , n ( a ) = f 1 ( a ) − R 1 , n ( a ) a n + 1 ,

    where R 1 , n ( a ) = − 1 + ∑ k = 1 n B 1 ( k ) a k and B 1 ( k ) ( k ≥ 1 ) is defined by (3.1). Then, G 1 , 2 n ( G 1 , ( 2 n + 1 ) ) is strictly increasing (decreasing) and concave (convex) on ( 0 , 1 / 2 ] with G 1 , n ( 0 + ) = B 1 ( n + 1 ) . In particular, the inequalities

    (3.6) 2 ( G 1 , 2 n ( 1 / 2 ) + 2 ζ ( 2 n + 1 ) ) a 2 n + 2 ≤ − b R ( a ) − R 1 , 2 n ( a ) + 2 ζ ( 2 n + 1 ) a 2 n + 1 ≤ ( G 1 , 2 n ( 1 / 2 ) + 2 ζ ( 2 n + 1 ) ) a 2 n + 1

    and

    (3.7) ( G 1 , ( 2 n + 1 ) ( 1 / 2 ) − 2 ζ ( 2 n + 1 ) ) a 2 n + 2 ≤ − b R ( a ) − R 1 , ( 2 n + 1 ) ( a ) − 2 ζ ( 2 n + 1 ) a 2 n + 2 ≤ 2 ( G 1 , ( 2 n + 1 ) ( 1 / 2 ) − 2 ζ ( 2 n + 1 ) ) a 2 n + 3

    hold for all n ∈ N and a ∈ ( 0 , 1 / 2 ] , with equality in each instance if and only if a = 1 / 2 .

Proof

(1) It follows from Lemma 2.1(2) and b = a ( 1 − a ) = [ 1 − ( 1 − 2 a ) 2 ] / 4 that

(3.8) f 1 ( a ) = − a ( 1 − a ) R ( a ) = − 1 − ( 1 − 2 a ) 2 4 log 16 + 4 ∑ n = 1 ∞ λ ( 2 n + 1 ) ( 1 − 2 a ) 2 n = − log 2 + ∑ n = 1 ∞ A 1 ( 2 n ) ( 1 − 2 a ) 2 n .

On the other hand, from Lemma 2.1(2), f 1 ( a ) can be expressed by

(3.9) f 1 ( a ) = − a ( 1 − a ) R ( a ) = − a ( 1 − a ) 1 a + 2 ∑ n = 1 ∞ ζ ( 2 n + 1 ) a 2 n = − 1 + ∑ n = 1 ∞ B 1 ( n ) a n .

(2) From Lemma 2.1(1), we know that A 1 ( 2 n ) > 0 for all n ≥ 2 . Differentiating f 1 gives

(3.10) f 1 ( k ) ( a ) = ∑ n = [ ( k + 1 ) / 2 ] ∞ ( − 1 ) k 2 k ( 2 n ) ! ( 2 n − k ) ! A 1 ( 2 n ) ( 1 − 2 a ) 2 n − k ,

where and in what follows [ ⋅ ] denotes the greatest integral function. Therefore, ( − 1 ) k f 1 ( k ) ( a ) ≥ 0 for all a ∈ ( 0 , 1 / 2 ] and k = 3 , 4 , … and f 1 ″ is decreasing on ( 0 , 1 / 2 ] due to f 1 ‴ ( a ) ≤ 0 . It follows from (3.9) that

f 1 ″ ( a ) = ∑ n = 2 ∞ n ( n − 1 ) B 1 ( n ) a n − 2

with f 1 ″ ( 0 + ) = 0 . Therefore, f 1 ″ ( a ) < 0 for a ∈ ( 0 , 1 / 2 ] , so that f 1 ′ ( a ) is strictly decreasing on ( 0 , 1 / 2 ] with f 1 ′ ( 1 / 2 ) = 0 , which leads to the desired monotonicity and concavity of f 1 .

(3) Differentiating

F 1 , n ( a ) = f 1 ( a ) − P 1 , n ( a ) ( 1 − 2 a ) 2 ( n + 1 ) = ∑ k = n + 1 ∞ A 1 ( 2 k ) ( 1 − 2 a ) 2 ( k − n − 1 ) = ∑ k = 0 ∞ A 1 ( 2 ( k + n + 1 ) ) ( 1 − 2 a ) 2 k

gives

(3.11) F 1 , n ( l ) ( a ) = ∑ k = [ ( l + 1 ) / 2 ] ∞ ( − 1 ) l 2 k ( 2 k ) ! ( 2 k − l ) ! A 1 ( 2 ( k + n + 1 ) ) ( 1 − 2 a ) 2 k − l ,

which implies that

F 1 , n ′ ( a ) = − 4 ∑ k = 1 ∞ k A 1 ( 2 ( k + n + 1 ) ) ( 1 − 2 a ) 2 k − 1

and

F 1 , n ″ ( a ) = 8 ∑ k = 1 ∞ k ( 2 k − 1 ) A 1 ( 2 ( k + n + 1 ) ) ( 1 − 2 a ) 2 k − 2 .

Therefore, the monotonicity and concavity properties of F 1 , n follow easily from A 1 ( 2 n ) > 0 for all n ≥ 2 . Clearly, F 1 , n ( 0 + ) = λ ( 2 n + 1 ) − 1 , F 1 , n ( 1 / 2 − ) = A 1 ( 2 n + 2 ) , the double inequality follows from the monotonicity and convexity properties of F 1 , n ( a ) .

(4) Let G 1 , n ( a ) = h 1 ( a ) / h 2 ( a ) , where h 2 ( a ) = a n + 1 and

(3.12) h 1 ( a ) = f 1 ( a ) − R 1 , n ( a ) .

Then h 1 ( m ) ( 0 + ) = h 2 ( m ) ( 0 + ) = 0 for all m ∈ N ∪ { 0 } with 0 ≤ m ≤ n and

(3.13) h 1 ( n + 1 ) ( a ) h 2 ( n + 1 ) ( a ) = f 1 ( n + 1 ) ( a ) ( n + 1 ) ! .

From Lemma 2.3, we know that the function G 1 , 2 n ( G 1 , ( 2 n + 1 ) ) has the same monotonicity with the function f ( 2 n + 1 ) ( f ( 2 n + 2 ) ) . It follows from ( − 1 ) n f 1 ( n ) ( a ) ≥ 0 for a ∈ ( 0 , 1 / 2 ] and n ≥ 3 that G 1 , 2 n ( G 1 , ( 2 n + 1 ) ) is increasing (decreasing) on ( 0 , 1 / 2 ] . Differentiating G 1 , n gives

( G 1 , n ( a ) ) ′ = h 1 ( a ) h 2 ( a ) ′ = a ( f 1 ( a ) − R 1 , n ( a ) ) ′ − ( n + 1 ) ( f 1 ( a ) − R 1 , n ( a ) ) a n + 2 .

Let

h 3 ( a ) = a ( f 1 ( a ) − R 1 , n ( a ) ) ′ − ( n + 1 ) ( f 1 ( a ) − R 1 , n ( a ) ) ,

and h 4 ( a ) = a n + 2 . Then from (3.9), we know that h 3 ( a ) can be rewritten as follows:

h 3 ( a ) = ∑ k = n + 1 ∞ ( k − n − 1 ) B 1 ( k ) a k ,

which leads to h 3 ( m ) ( 0 + ) = h 4 ( m ) ( 0 + ) = 0 for all m ∈ N ∪ { 0 } with 0 ≤ m ≤ n and

(3.14) h 3 ( n + 1 ) ( a ) h 4 ( n + 1 ) ( a ) = f 1 ( n + 2 ) ( a ) ( n + 2 ) ! .

From Lemma 2.3, we know that the function ( G 1 , n ) ′ has the same monotonicity with the function f ( n + 2 ) . Therefore, the desired monotonicity of ( G 1 , 2 n ) ′ and ( G 1 , ( 2 n + 1 ) ) ′ is obtained, the concavity (convexity) property of G 1 , 2 n ( G 1 , ( 2 n + 1 ) ) follows from part (2), and the double inequality (3.6) (3.7) and its equality case follow from the monotonicity and concavity (convexity) properties of G 1 , 2 n ( G 1 , ( 2 n + 1 ) ) .□

A function f is said to be strictly completely monotonic on an interval I ⊂ R if ( − 1 ) n f ( n ) ( x ) > 0 for all x ∈ I and n = 0 , 1 , 2 , 3 … . If ( − 1 ) n f ( n ) ( x ) ≥ 0 for all x ∈ I and n = 0 , 1 , 2 , 3 … , then f is called completely monotonic on I . The completely monotonic functions have important applications in areas of numerical analysis [64,65], probability theory [66], physics [67], and so on.

Corollary 3.2

For each n ∈ N , the function F 1 , n is completely monotonic on ( 0 , 1 / 2 ] .

Proof

It follows from (3.11) that ( − 1 ) l F 1 , n ( l ) ( a ) ≥ 0 for all a ∈ ( 0 , 1 / 2 ] , and l = 0 , 1 , 2 , 3 … . Therefore, F 1 , n is completely monotonic on ( 0 , 1 / 2 ] .□

Let

(3.15) A 2 ( 2 ) = [ 9 λ ( 3 ) − 10 log 2 ] / 4 , A 2 ( 4 ) = [ 9 λ ( 5 ) − 10 λ ( 3 ) + log 2 ] / 4 , A 2 ( 2 k ) = [ 9 λ ( 2 k + 1 ) − 10 λ ( 2 k − 1 ) + λ ( 2 k − 3 ) ] / 4 ( k ≥ 3 ) , B 2 ( 1 ) = − 1 , B 2 ( 2 ) = − 2 , B 2 ( 3 ) = 4 ζ ( 3 ) + 1 , B 2 ( 4 ) = − 2 ζ ( 3 ) , B 2 ( 2 k + 1 ) = − 4 ( ζ ( 2 k − 1 ) − ζ ( 2 k + 1 ) ) ( k ≥ 2 ) , B 2 ( 2 k + 2 ) = 2 ( ζ ( 2 k − 1 ) − ζ ( 2 k + 1 ) ) ( k ≥ 2 ) .

Theorem 3.3

Let f 2 ( a ) = b ( 2 + b ) R ( a ) . Then, the following statements are true:

  1. f 2 has the power series formula:

    (3.16) f 2 ( a ) = 9 log 2 4 + ∑ n = 1 ∞ A 2 ( 2 n ) ( 1 − 2 a ) 2 n = 2 + ∑ n = 1 ∞ B 2 ( n ) a n .

  2. The function f 2 is strictly decreasing on ( 0 , 1 / 2 ] , and f 2 is neither convex nor concave on ( 0 , 1 / 2 ] and ( − 1 ) k f 2 ( k ) ( a ) > 0 for ( 0 , 1 / 2 ] and k ≥ 5 .

  3. Let

    (3.17) F 2 , n ( a ) = f 2 ( a ) − P 2 , n ( a ) ( 1 − 2 a ) 2 ( n + 1 ) ,

    where P 2 , n ( a ) = ( 9 log 2 ) / 4 + ∑ k = 1 n A 2 ( 2 k ) ( 1 − 2 a ) 2 k and A 2 ( 2 k ) ( k ≥ 1 ) is given in (3.15). Then, F 2 , n is strictly decreasing and convex from ( 0 , 1 / 2 ) onto ( A 2 ( n + 1 ) , F 2 , n ( 0 + ) ) . In particular, the double inequality

    (3.18) 0 ≤ b ( 2 + b ) R ( a ) − P 2 , n ( a ) − A 2 ( n + 1 ) ( 1 − 2 a ) 2 ( n + 1 ) ≤ [ F 2 , n ( 0 + ) − A 2 ( n + 1 ) ] ( 1 − 2 a ) 2 ( n + 1 ) + 1

    holds for all n ∈ N and a ∈ ( 0 , 1 / 2 ] , with equality in each instance if and only if a = 1 / 2 .

  4. Let

    (3.19) G 2 , n ( a ) = f 2 ( a ) − R 2 , n ( a ) a n + 1 ,

    where R 2 , n ( a ) = 2 + ∑ k = 1 n B 2 ( k ) a k and B 2 ( k ) ( k ≥ 1 ) is defined by (3.15). Then, for n ≥ 2 , G 2 , 2 n ( G 2 , ( 2 n − 1 ) ) is strictly increasing (decreasing) and concave (convex) on ( 0 , 1 / 2 ] with G 2 , n ( 0 + ) = B 2 ( n + 1 ) . In particular, inequalities

    (3.20) 2 ( G 2 , 2 n ( 1 / 2 ) − B 2 ( 2 n + 1 ) ) a 2 n + 2 ≤ b ( 2 + b ) R ( a ) − R 2 , 2 n ( a ) − B 2 ( 2 n + 1 ) a 2 n + 1 ≤ ( G 2 , 2 n ( 1 / 2 ) − B 2 ( 2 n + 1 ) ) a 2 n + 1

    and

    (3.21) ( G 2 , ( 2 n − 1 ) ( 1 / 2 ) − B 2 ( 2 n ) ) a 2 n + 2 ≤ b ( 2 + b ) R ( a ) − R 2 , 2 n ( a ) − B 2 ( 2 n ) a 2 n + 2 ≤ 2 ( G 2 , ( 2 n − 1 ) ( 1 / 2 ) − B 2 ( 2 n ) ) a 2 n + 3

    hold for all n ∈ N and a ∈ ( 0 , 1 / 2 ] , with equality in each instance if and only if a = 1 / 2 .

Proof

(1) It follows from Lemma 2.1(2) and b = a ( 1 − a ) = [ 1 − ( 1 − 2 a ) 2 ] / 4 that

(3.22) f 2 ( a ) = b ( 2 + b ) R ( a ) = 9 − 10 ( 1 − 2 a ) 2 + ( 1 − 2 a ) 4 16 log 16 + 4 ∑ n = 1 ∞ λ ( 2 n + 1 ) ( 1 − 2 a ) 2 n = 9 log 2 4 + ∑ n = 1 ∞ A 2 ( 2 n ) ( 1 − 2 a ) 2 n ,

where A 2 ( 2 n ) ( n ≥ 1 ) is defined by (3.15). Again use Lemma 2.1(2), we obtain

(3.23) f 2 ( a ) = b ( 2 + b ) R ( a ) = b ( 2 + b ) 1 a + 2 ∑ n = 1 ∞ ζ ( 2 n + 1 ) a 2 n = 2 − a − 2 a 2 + ( 4 ζ ( 3 ) + 1 ) a 3 − 2 ζ ( 3 ) a 4 + ∑ n = 2 ∞ [ − 4 ζ ( 2 n − 1 ) + 4 ζ ( 2 n + 1 ) ] a 2 n + 1 + [ 2 ζ ( 2 n − 1 ) − 2 ζ ( 2 n + 1 ) ] a 2 n + 2 = 2 + ∑ n = 1 ∞ B 2 ( n ) a n .

(2) It follows from (1.16) that A 2 ( 2 n ) = [ 9 λ ( 2 n + 1 ) − 10 λ ( 2 n − 1 ) + λ ( 2 n − 3 ) ] / 4 and

A 2 ( 2 n ) = 1 4 ∑ k = 1 ∞ 9 − 10 ( 2 k + 1 ) 2 + ( 2 k + 1 ) 4 ( 2 k + 1 ) 2 n + 1 = 4 ∑ k = 2 n ( k 2 + 2 k ) ( k 2 − 1 ) ( 2 k + 1 ) 2 n + 1 > 0

for n > 2 . Differentiating f 2 gives

(3.24) f 2 ( k ) ( a ) = ∑ n = [ ( k + 1 ) / 2 ] ∞ ( − 1 ) k 2 k ( 2 n ) ! ( 2 n − k ) ! A 2 ( 2 n ) ( 1 − 2 a ) 2 n − k ,

(3.25) f 2 ( k ) ( a ) = ∑ n = [ ( k + 1 ) / 2 ] ∞ ( n ) ! ( n − k ) ! B 2 ( n ) a n − k .

From (3.24), we know that ( − 1 ) k f 2 ( k ) ( a ) ≥ 0 for a ∈ ( 0 , 1 / 2 ] and k = 5 , 6 , … ; and f 2 ( 5 ) ( a ) is increasing on ( 0 , 1 / 2 ] . By the series formula of f 2 ( a ) , we obtain f 2 ( 5 ) ( 1 / 2 ) = 0 ; therefore, f 2 ( 5 ) ( a ) ≤ 0 for a ∈ ( 0 , 1 / 2 ] , so that f 2 ( 4 ) is strictly decreasing on ( 0 , 1 / 2 ] .

It follows from (3.25) that f 2 ( 4 ) ( 0 + ) = − 48 ζ ( 3 ) < 0 , so f 2 ( 4 ) ( a ) < 0 for a ∈ ( 0 , 1 / 2 ] . Hence, f 2 ‴ ( a ) are strictly decreasing on ( 0 , 1 / 2 ] with f 2 ‴ ( 1 / 2 ) = 0 , which implies that f 2 ‴ ( a ) > 0 and f 2 ″ are strictly increasing on ( 0 , 1 / 2 ] .

By (3.24) and (3.25), we obtain f 2 ″ ( 0 + ) = − 4 and f 2 ″ ( 1 / 2 ) = − 20 log 2 + 63 ζ ( 3 ) / 4 = 5.070 … > 0 . Therefore, there exists a ∗ ∈ ( 0 , 1 / 2 ] such that f 2 ″ ( a ∗ ) = 0 , and f 2 ′ is decreasing on ( 0 , a ∗ ] and f 2 ′ is increasing on on ( a ∗ , 1 / 2 ] . Since f 2 ′ ( 0 + ) = − 1 and f 2 ′ ( 1 / 2 ) = 0 , so f 2 ′ ( a ) < 0 and f 2 is strictly decreasing on ( 0 , 1 / 2 ] with f 2 ( 0 + ) = 2 and f 2 ( 1 / 2 ) = 9 log 2 / 4 .

(3) Differentiating

F 2 , n ( a ) = f 2 ( a ) − P 2 , n ( a ) ( 1 − 2 a ) 2 ( n + 1 ) = ∑ k = n + 1 ∞ A 2 ( 2 k ) ( 1 − 2 a ) 2 ( k − n − 1 ) = ∑ k = 0 ∞ A 2 ( 2 ( k + n + 1 ) ) ( 1 − 2 a ) 2 k

leads to

(3.26) F 2 , n ( l ) ( a ) = ∑ k = [ ( l + 1 / 2 ) ] ∞ ( − 1 ) l 2 k ( 2 k ) ! ( 2 k − l ) ! A 2 ( 2 ( k + n + 1 ) ) ( 1 − 2 a ) 2 k − l ,

which implies that

F 2 , n ′ ( a ) = − 4 ∑ k = 1 ∞ k A 2 ( 2 ( k + n + 1 ) ) ( 1 − 2 a ) 2 k − 1

and

F 2 , n ″ ( a ) = 8 ∑ k = 1 ∞ k ( 2 k − 1 ) A 2 ( 2 ( k + n + 1 ) ) ( 1 − 2 a ) 2 k − 2 .

Therefore, the monotonicity and concavity properties of F 2 , n follow from A 2 ( 2 n ) > 0 for all n > 2 , and the desired inequalities follow from the monotonicity and convexity properties of F 2 , n together with F 2 , n ( ( 1 / 2 ) − ) = A 2 ( 2 n + 2 ) .

(4) Let G 2 , n ( a ) = h 5 ( a ) / h 6 ( a ) , where h 6 ( a ) = a n + 1 and

(3.27) h 5 ( a ) = f 2 ( a ) − R 2 n ( a ) .

Then it is easy to verify that h 5 ( m ) ( 0 + ) = h 6 ( m ) ( 0 + ) = 0 for all m ∈ N ∪ { 0 } with 0 ≤ m ≤ n . Simple computations lead to

(3.28) h 5 ( n + 1 ) ( a ) h 6 ( n + 1 ) ( a ) = f 2 ( n + 1 ) ( a ) ( n + 1 ) ! .

From Lemma 2.3, we know that G 2 , 2 n ( G 2 , ( 2 n − 1 ) ) has the same monotonicity with the function f 2 ( 2 n + 1 ) ( f 2 ( 2 n ) ) . Therefore, G 2 , 2 n ( G 2 , ( 2 n − 1 ) ) is increasing (decreasing) on ( 0 , 1 / 2 ] due to ( − 1 ) n f 2 ( n ) ( a ) ≥ 0 for a ∈ ( 0 , 1 / 2 ] and n ≥ 5 .

Differentiating G 2 , n gives

( G 2 , n ( a ) ) ′ = h 5 ( a ) h 6 ( a ) ′ = a ( f 2 ( a ) − R 2 , n ( a ) ) ′ − ( n + 1 ) ( f 2 ( a ) − R 2 , n ( a ) ) a n + 2 .

Let h 8 ( a ) = a n + 2 and

h 7 ( a ) = a ( f 2 ( a ) − R 2 , n ( a ) ) ′ − ( n + 1 ) ( f 2 ( a ) − R 2 , n ( a ) ) .

Then from (3.23), we know that h 7 ( a ) can be rewritten as follows:

h 7 ( a ) = ∑ k = n + 1 ∞ ( k − n − 1 ) B 2 ( k ) a k .

Therefore, h 7 ( m ) ( 0 + ) = h 8 ( m ) ( 0 + ) = 0 for all m ∈ N ∪ { 0 } with 0 ≤ m ≤ n and

(3.29) h 7 ( n + 1 ) ( a ) h 8 ( n + 1 ) ( a ) = f 2 ( n + 2 ) ( a ) ( n + 2 ) ! ,

which shows that G 2 , n ′ has the same monotonicity with the function f 2 ( n + 2 ) by Lemma 2.3. Therefore, the desired monotonicity of G 2 , 2 n ′ and G 2 , ( 2 n − 1 ) ′ , the concavity (convexity) of G 2 , 2 n ( G 2 , ( 2 n − 1 ) ) , are obtained, and the inequalities (3.20) and (3.21) follow easily from the monotonicity and concavity (convexity) properties of G 2 , 2 n ( G 2 , ( 2 n + 1 ) ) .□

Corollary 3.4

The function F 2 , n ( n > 2 ) is completely monotonic on ( 0 , 1 / 2 ] .

Proof

From (3.26), we clearly see that A 2 ( 2 n ) > 0 for all n > 2 . Therefore, ( − 1 ) l F 2 , n ( l ) ( a ) ≥ 0 for all a ∈ ( 0 , 1 / 2 ] and l = 0 , 1 , 2 , 3 … , and F 2 , n is completely monotonic on ( 0 , 1 / 2 ] .□

Corollary 3.5

G 2 , 1 is increasing and concave, and G 2 , 2 is decreasing and concave on ( 0 , 1 / 2 ] .

Proof

It follows from (3.28) and Lemma 2.3 that G 2 , 2 ( G 2 , 1 ) has the same monotonicity with the function f 2 ‴ ( f 2 ″ ) . Since f 2 ‴ ( f 2 ″ ) is strictly decreasing (increasing) on ( 0 , 1 / 2 ] , we can obtain the desired monotonicity of G 2 , 2 and G 2 , 1 . It follows from (3.29) and Lemma 2.3 that G 2 , 2 ′ ( G 2 , 1 ′ ) has the same monotonicity with the function f 2 ( 4 ) ( f 2 ‴ ) . Since f 2 ‴ and f 2 ( 4 ) is strictly decreasing on ( 0 , 1 / 2 ] , the desired monotonicity of G 2 , 2 ′ ( a ) and G 2 , 1 ′ ( a ) , and the concavity (convexity) of G 2 , 2 ( a ) and G 2 , 1 ( a ) are obtained.□

Let

(3.30) A 3 ( 0 ) = 7 log 2 − 4 = 0.8520 … , A 3 ( 2 ) = 7 λ ( 3 ) + log 2 − 4 = 4.055745 … , A 3 ( 2 k ) = 7 λ ( 2 k + 1 ) + λ ( 2 k − 1 ) − 4 ( k ≥ 2 ) , B 3 ( 1 ) = 0 , B 3 ( 2 ) = 4 ζ ( 3 ) − 1 , B 3 ( 2 k − 1 ) = − 1 − 2 ζ ( 2 k − 1 ) ( k ≥ 2 ) , B 3 ( 2 k ) = − 1 + 2 ζ ( 2 k − 1 ) + 4 ζ ( 2 k + 1 ) ( k ≥ 2 ) .

Theorem 3.6

Let f 3 ( a ) = ( 2 − b ) R ( a ) − 1 / b . Then the following statements are true:

  1. f 3 has the power series formula:

    (3.31) f 3 ( a ) = 7 log 2 − 4 + ∑ n = 1 ∞ A 3 ( 2 n ) ( 1 − 2 a ) 2 n = 1 a − 2 + ∑ n = 1 ∞ B 3 ( n ) a n .

  2. f 3 ( a ) is completely monotonic on ( 0 , 1 / 2 ] .

  3. Let

    (3.32) F 3 , n ( a ) = f 3 ( a ) − P 3 , n ( a ) ( 1 − 2 a ) 2 ( n + 1 ) ,

    where P 3 , n ( a ) = ∑ k = 0 n A 3 ( 2 k ) ( 1 − 2 a ) 2 k and A 3 ( k ) ( k ≥ 0 ) is given in (3.30). Then, F 3 , n is strictly decreasing and convex from ( 0 , 1 / 2 ] onto ( A 3 ( n + 1 ) , ∞ ) . In particular, the inequality

    (3.33) A 3 ( n + 1 ) ( 1 − 2 a ) 2 ( n + 1 ) < ( 2 − b ) R − 1 / b − P 3 , n ( a )

    holds for all a ∈ ( 0 , 1 / 2 ] and n ∈ N , with equality in each instance if and only if a = 1 / 2 .

  4. Let

    (3.34) G 3 , n ( a ) = f 3 ( a ) − R 3 , 2 n ( a ) a 2 n + 1 ,

    where R 3 , n ( a ) = 1 / a − 2 + ∑ k = 1 n B 3 ( k ) a k and B 3 ( k ) ( k ≥ 1 ) is defined in (3.30). Then, G 3 , n is strictly increasing from ( 0 , 1 / 2 ] onto ( B 3 ( 2 n − 1 ) , G 3 , n ( 1 / 2 ) ] . In particular, the inequality

    (3.35) B 3 ( 2 n − 1 ) a 2 n − 1 < ( 2 − b ) R ( a ) − 1 / b − R 3 , n ( a ) ≤ G 3 , n 1 2 a 2 n − 1 ,

    holds for all a ∈ ( 0 , 1 / 2 ] and n ∈ N , with equality in each instance if and only if a = 1 / 2 .

Let

(3.36) H 3 , n ( a ) = f 3 ( a ) − R 3 , ( 2 n − 1 ) ( a ) a 2 n ,

where R 3 , n ( a ) = 1 / a − 2 + ∑ k = 1 n B 3 ( k ) a k and B 3 ( k ) ( k ≥ 1 ) is given in (3.30). Then, H 3 , n is convex on ( 0 , 1 / 2 ] and there exists a 0 ∈ ( 0 , 1 / 2 ] such that H 3 , n is strictly decreasing on ( 0 , a 0 ] and strictly increasing on ( a 0 , 1 / 2 ] . In particular, the inequality

(3.37) ( 2 − b ) R ( a ) − 1 / b − R 3 , ( 2 n − 1 ) ( a ) < min B 3 ( 2 n ) a 2 n − 1 , H 3 , n 1 2 a 2 n − 1

holds for a ∈ ( 0 , 1 / 2 ] and n ∈ N .

Proof

(1) It follows from Lemma 2.1(2) that

(3.38) f 3 ( a ) = ( 2 − b ) R − 1 b = 7 + ( 1 − 2 a ) 2 4 R − 4 1 − ( 1 − 2 a ) 2 = [ 7 + ( 1 − 2 a ) 2 ] log 2 + ∑ k = 1 ∞ λ ( 2 k + 1 ) ( 1 − 2 a ) 2 k − 4 ∑ k = 0 ∞ ( 1 − 2 a ) 2 k = 7 log 2 − 4 + ∑ n = 1 ∞ A 3 ( 2 n ) ( 1 − 2 a ) 2 n ,

where A 3 ( 2 n ) ( n ≥ 1 ) is given by (3.30), which leads to

(3.39) f 3 ( a ) = ( 2 − b ) R − 1 b = [ 2 − a ( 1 − a ) ] 1 a + 2 ∑ n = 1 ∞ ζ ( 2 n + 1 ) a 2 n − 1 a ( 1 − a ) = 2 a − ( 1 − a ) + 2 ( 2 − a + a 2 ) ∑ n = 1 ∞ ζ ( 2 n + 1 ) a 2 n − 1 a + ∑ n = 0 ∞ a n = 1 a − 2 + ( 4 ζ ( 3 ) − 1 ) a 2 + ∑ n = 2 ∞ [ ( − 1 − 2 ζ ( 2 n − 1 ) ) a 2 n − 1 + ( − 1 + 2 ζ ( 2 n − 1 ) + 4 ζ ( 2 n + 1 ) ) a 2 n ] = 1 a − 2 + ∑ n = 1 ∞ B 3 ( n ) a n ,

where B 3 ( n ) ( n ≥ 2 ) is defined by (3.30).

(2) From Lemma 2.1(1), we know that 7 log 2 − 4 = 0.8520 … , 7 λ ( 3 ) + log 2 − 4 = 4.055745 … and A 3 ( n ) > 4 for all n ∈ N . Simple computations lead to

(3.40) f 3 ( k ) ( a ) = ∑ n = [ ( k + 1 ) / 2 ] ∞ ( − 1 ) k 2 k ( 2 n ) ! ( 2 n − k ) ! A 3 ( 2 n ) ( 1 − 2 a ) 2 n − k ,

therefore, ( − 1 ) k f 3 ( k ) ( a ) ≥ 0 for all a ∈ ( 0 , 1 / 2 ] and k = 0 , 1 , 2 , … , so f 3 is completely monotonic on ( 0 , 1 / 2 ] .

(3) Differentiating

F 3 , n ( a ) = f 3 ( a ) − P 3 , n ( a ) ( 1 − 2 a ) 2 ( n + 1 ) = ∑ k = n + 1 ∞ A 3 ( 2 k ) ( 1 − 2 a ) 2 ( k − n − 1 ) = ∑ k = 0 ∞ A 3 ( 2 ( k + n + 1 ) ) ( 1 − 2 a ) 2 k

gives

(3.41) F 3 , n ( l ) ( a ) = ∑ k = [ ( l + 1 ) / 2 ] ∞ ( − 1 ) l 2 k ( 2 k ) ! ( 2 k − l ) ! A 3 ( 2 ( k + n + 1 ) ) ( 1 − 2 a ) 2 k − l ,

which implies

F 3 , n ′ ( a ) = − 4 ∑ k = 1 ∞ k A 3 ( k + n + 1 ) ( 1 − 2 a ) 2 k − 1

and

F 3 , n ″ ( a ) = 8 ∑ k = 1 ∞ k ( 2 k − 1 ) A 3 ( k + n + 1 ) ( 1 − 2 a ) 2 k − 2 .

From Lemma 2.1(1), we know that 7 λ ( 3 ) + log 2 − 4 = 4.055745 … and A 3 ( 2 n ) > 4 for all n ∈ N . Therefore, we obtain the desired monotonicity and concavity of F 3 , n , and inequality (3.33) follows from the monotonicity of F 3 , n and F 3 , n ( 1 / 2 ) = A 3 ( n + 1 ) .

(4) Differentiating

(3.42) G 3 , n ( a ) = f 3 ( a ) − R 3 , ( 2 n ) ( a ) a 2 n + 1 = ∑ k = 2 n + 1 ∞ B 3 ( k ) a k − ( 2 n + 1 ) = ∑ k = 0 ∞ [ B 3 ( 2 k + 2 n + 1 ) a 2 k + B 3 ( 2 k + 2 n + 2 ) a 2 k + 1 ]

gives

G 3 , n ′ ( a ) = ∑ k = 0 ∞ [ ( 2 k + 1 ) B 3 ( 2 k + 2 n + 2 ) a 2 k + ( 2 k + 2 ) B 3 ( 2 k + 2 n + 3 ) a 2 k + 1 ] = ∑ k = 0 ∞ [ ( 2 k + 1 ) B 3 ( 2 k + 2 n + 2 ) + ( 2 k + 2 ) B 3 ( 2 k + 2 n + 3 ) a ] a 2 k .

Note that ζ ( 2 n + 1 ) is decreasing, 0 < ζ ( 2 n + 1 ) < 2 , B 3 ( 2 k + 2 n + 2 ) > 5 , and B 3 ( 2 k + 2 n + 3 ) > − 5 . Let M 1 ( a ) = ( 2 k + 1 ) B 3 ( 2 k + 2 n + 2 ) + ( 2 k + 2 ) B 3 ( 2 k + 2 n + 3 ) a . Then, M 1 is strictly decreasing on ( 0 , 1 / 2 ] , M 1 ( a ) ≥ M 1 ( 1 / 2 ) > 5 ( 2 k + 1 ) − 5 ( k + 1 ) = 5 k ≥ 0 for n ∈ N , k ≥ 0 and a ∈ ( 0 , 1 / 2 ] . Therefore, G 3 , n ′ ( a ) > 0 and G 3 , n is strictly increasing on ( 0 , 1 / 2 ] . Clearly, G 3 , n ( 0 + ) = B 3 ( 2 n − 1 ) .

From (3.39), we obtain

(3.43) H 3 , n ( a ) = f 3 ( a ) − R 3 , ( 2 n − 1 ) ( a ) a 2 n = ∑ k = n ∞ [ B 3 ( 2 k ) a 2 ( k − n ) + B 3 ( 2 k + 1 ) a 2 ( k − n ) + 1 ] = ∑ k = 0 ∞ [ B 3 ( 2 k + 2 n ) a 2 k + B 3 ( 2 k + 2 n + 1 ) a 2 k + 1 ] .

Differentiating H 3 , n yields

H 3 , n ′ ( a ) = B 3 ( 2 n + 1 ) + ∑ k = 1 ∞ [ 2 k B 3 ( 2 k + 2 n ) a 2 k − 1 + ( 2 k + 1 ) B 3 ( 2 k + 2 n + 1 ) a 2 k ] = B 3 ( 2 n + 1 ) + ∑ k = 1 ∞ [ 2 k B 3 ( 2 k + 2 n ) + ( 2 k + 1 ) B 3 ( 2 k + 2 n + 1 ) a ] a 2 k − 1

and

H 3 , n ″ ( a ) = ∑ k = 1 ∞ [ 2 k ( 2 k − 1 ) B 3 ( 2 k + 2 n ) a 2 k − 2 + 2 k ( 2 k + 1 ) B 3 ( 2 k + 2 n + 1 ) a 2 k − 1 ] = ∑ k = 1 ∞ [ 2 k ( 2 k − 1 ) B 3 ( 2 k + 2 n ) + 2 k ( 2 k + 1 ) B 3 ( 2 k + 2 n + 1 ) a ] a 2 k − 2 .

Note that ζ ( 2 n + 1 ) is decreasing, 0 < ζ ( 2 n + 1 ) < 2 , B 3 ( 2 k + 2 n ) > 5 and B 3 ( 2 k + 2 n + 1 ) > − 5 . Let M 2 ( a ) = 2 k B 3 ( 2 k + 2 n ) + ( 2 k + 1 ) B 3 ( 2 k + 2 n + 1 ) a . Then, M 2 ( a ) is strictly decreasing on ( 0 , 1 / 2 ] and M 2 ( a ) ≥ M 2 ( 1 / 2 ) > 10 k − 5 ( 2 k + 1 ) / 2 = 5 k − 5 / 2 > 0 for k ≥ 1 and a ∈ ( 0 , 1 / 2 ] . Let M 3 ( a ) = 2 k ( 2 k − 1 ) B 3 ( 2 k + 2 n ) + 2 k ( 2 k + 1 ) B 3 ( 2 k + 2 n + 1 ) a . Then, M 3 ( a ) is strictly decreasing on ( 0 , 1 / 2 ] .

If k = 1 and a ∈ ( 0 , 1 / 2 ] , then we obtain

2 B 3 ( 2 + 2 n ) + 6 B 3 ( 2 n + 3 ) a ≥ 2 B 3 ( 2 + 2 n ) + 3 B 3 ( 2 n + 3 ) = − 5 + 4 ζ ( 2 n + 1 ) + 2 ζ ( 2 n + 3 ) > 1 .

If k ≥ 2 and a ∈ ( 0 , 1 / 2 ] , then one has

M 3 ( a ) ≥ M 3 ( 1 / 2 ) > 10 k ( 2 k − 1 ) − 5 k ( 2 k + 1 ) = 5 k ( 2 k − 3 ) > 0 .

Therefore, M 3 ( a ) > 0 for k ≥ 1 and a ∈ ( 0 , 1 / 2 ] , H 3 , n ″ ( a ) > 0 , H 3 , n ′ is strictly increasing on ( 0 , 1 / 2 ] , and H 3 , n is convex on ( 0 , 1 / 2 ] . Clearly, H 3 , n ′ ( 0 + ) = B 3 ( 2 n + 1 ) < 0 and

H 3 , n ′ 1 2 = B 3 ( 2 n + 1 ) + ∑ k = 1 ∞ 2 k B 3 ( 2 k + 2 n ) + ( 2 k + 1 ) B 3 ( 2 k + 2 n + 1 ) 2 1 2 2 k − 1 > B 3 ( 2 n + 1 ) + ∑ k = 1 2 2 k B 3 ( 2 k + 2 n ) + ( 2 k + 1 ) B 3 ( 2 k + 2 n + 1 ) 2 1 2 2 k − 1 = − 57 16 + 7 ζ ( 2 n + 3 ) 2 + 11 ζ ( 2 n + 5 ) 8 > 21 11 ,

which leads to the conclusion that there exists a 0 ∈ ( 0 , 1 / 2 ] such that H 3 , n is decreasing on ( 0 , a 0 ] and increasing on [ a 0 , 1 / 2 ] .□

Let

(3.44) A 4 ( 0 ) = 4 log 2 − 15 / 4 = − 0.9774 … , A 4 ( 2 ) = 4 λ ( 3 ) − 17 / 4 = − 0.0426 … , A 4 ( 2 k ) = 4 ( λ ( 2 k + 1 ) − 1 ) ( k ≥ 2 ) , B 4 ( 1 ) = 0 , B 4 ( 2 ) = 2 ζ ( 3 ) − 2 , B 4 ( 2 k − 1 ) = − 1 , B 4 ( 2 k ) = 2 ζ ( 2 k + 1 ) − 1 ( k ≥ 2 ) .

Theorem 3.7

Let f 4 ( a ) = R ( a ) + b − 1 / b . Then the following statements are true:

  1. f 4 has the power series formula

    (3.45) f 4 ( a ) = 4 log 2 − 15 4 + ∑ n = 1 ∞ A 4 ( 2 n ) ( 1 − 2 a ) 2 n = − 1 + ∑ k = 1 ∞ B 4 ( k ) a k .

  2. The function f 4 ( a ) is strictly increasing on ( 0 , 1 / 2 ] , and f 4 is neither convex nor concave on ( 0 , 1 / 2 ] and ( − 1 ) k f 4 ( k ) ( a ) > 0 for k ≥ 3 and a ∈ ( 0 , 1 / 2 ] .

  3. Let

    (3.46) F 4 , n ( a ) = f 4 ( a ) − P 4 , n ( a ) ( 1 − 2 a ) 2 ( n + 1 ) ,

    where P 4 , n ( a ) = ∑ k = 0 n A 4 ( 2 k ) ( 1 − 2 a ) 2 k and A 4 ( 2 k ) ( k ≥ 0 ) is defined in (3.44). Then F 4 , n is strictly decreasing and convex from ( 0 , 1 / 2 ] onto [ A 4 ( 2 n + 2 ) , F 4 , n ( 0 + ) ) , and the inequality

    (3.47) 0 ≤ R ( a ) + b − 1 b − P 4 , n ( a ) − A 4 ( 2 n + 2 ) ( 1 − 2 a ) 2 ( n + 1 ) ≤ ( F 4 , n ( 0 + ) − A 4 ( 2 n + 2 ) ) ( 1 − 2 a ) 2 n + 3

    holds for all n ∈ N and a ∈ ( 0 , 1 / 2 ] , with equality in each instance if and only if a = 1 / 2 .

  4. Let

    (3.48) G 4 , n ( a ) = f 4 ( a ) − R 4 , n ( a ) a n + 1 ,

    where R 4 , n ( a ) = − 1 + ∑ k = 1 n B 4 ( k ) a k and B 4 ( k ) ( k ≥ 1 ) is given by (3.44). Then G 2 , 2 n ( G 2 , ( 2 n − 1 ) ) is strictly increasing (decreasing) and concave (convex) on ( 0 , 1 / 2 ] for n ≥ 2 with G 4 n ( 0 + ) = B 4 ( n + 1 ) . In particular, the inequalities

    (3.49) 2 ( G 4 , 2 n ( 1 / 2 ) − B 4 ( 2 n + 1 ) ) a 2 n + 2 ≤ R ( a ) + b − 1 b − R 4 , 2 n ( a ) − B 4 ( 2 n + 1 ) a 2 n + 1 ≤ ( G 4 , 2 n ( 1 / 2 ) − B 4 ( 2 n + 1 ) ) a 2 n + 1

    and

    (3.50) ( G 4 , ( 2 n − 1 ) ( 1 / 2 ) − B 4 ( 2 n ) ) a 2 n + 2 ≤ R ( a ) + b − 1 b − R 4 , 2 n ( a ) − B 4 ( 2 n ) a 2 n + 2 ≤ 2 ( G 4 , ( 2 n − 1 ) ( 1 / 2 ) − B 4 ( 2 n ) ) a 2 n + 3

    hold for all n ∈ N and a ∈ ( 0 , 1 / 2 ] , with equality in each instance if and only if a = 1 / 2 .

Proof

(1) It follows from Lemma 2.1(2) that

(3.51) f 4 ( a ) = R ( a ) + b − 1 b = − 1 b + b + 1 b − 4 ( 1 − log 2 ) + 4 ∑ n = 1 ∞ [ λ ( 2 n + 1 ) − 1 ] ( 1 − 2 a ) 2 n = 1 4 − ( 1 − 2 a ) 2 4 − 4 ( 1 − log 2 ) + 4 ∑ n = 1 ∞ [ λ ( 2 n + 1 ) − 1 ] ( 1 − 2 a ) 2 n = 4 log 2 − 15 4 + ∑ n = 1 ∞ A 4 ( 2 n ) ( 1 − 2 a ) 2 n ,

where A 4 ( n ) ( n ≥ 1 ) is given in (3.44). Therefore,

(3.52) f 4 ( a ) = R ( a ) + b − 1 b = 1 a + ∑ n = 1 ∞ 2 ζ ( 2 n + 1 ) a 2 n + a ( 1 − a ) − 1 a ( 1 − a ) = 1 a + ∑ n = 1 ∞ 2 ζ ( 2 n + 1 ) a 2 n − 1 a + 1 + a + a 2 + ∑ n = 3 ∞ a n + a − a 2 = − 1 − ( 2 − 2 ζ ( 3 ) ) a 2 − ∑ n = 2 ∞ [ a 2 n − 1 + ( 1 − 2 ζ ( 2 n + 1 ) ) a 2 n ] = − 1 + ∑ k = 1 ∞ B 4 ( k ) a k .

(2) It follows from Lemma 2.1(1) that A 4 ( 2 n ) = 4 ( λ ( 2 k + 1 ) − 1 ) > 0 for all n ≥ 2 and n ∈ N . Elaborated computations lead to

(3.53) f 4 ( k ) ( a ) = ∑ n = [ ( k + 1 ) / 2 ] ∞ ( − 1 ) k 2 k ( 2 n ) ! ( 2 n − k ) ! A 4 ( 2 n ) ( 1 − 2 a ) 2 n − k

and

(3.54) f 4 ( k ) ( a ) = ∑ n = [ ( k + 1 ) / 2 ] ∞ ( n ) ! ( n − k ) ! B 4 ( n ) a n − k ,

which imply that ( − 1 ) k f 4 ( k ) ( a ) ≥ 0 for all a ∈ ( 0 , 1 / 2 ] and k = 3 , 4 , … . Therefore, f 4 ‴ ( a ) ≤ 0 and f 4 ″ ( a ) are decreasing on ( 0 , 1 / 2 ] . Note that f 4 ″ ( a ) can be written as follows:

f 4 ″ ( a ) = ∑ n = 2 ∞ n ( n − 1 ) B 4 ( n ) a n − 2 .

It is easy to check that f 4 ″ ( 0 + ) = 4 ζ ( 3 ) − 4 > 0 and f 4 ″ ( 1 / 2 ) = − 34 + 28 ζ ( 3 ) = − 0.341 … . Therefore, there exists a 1 ∈ ( 0 , 1 / 2 ] such that f 4 ″ ( a ) ≥ 0 on ( 0 , a 1 ] and f 4 ″ ( a ) < 0 on ( a 1 , 1 / 2 ] , and f ′ ( a ) is increasing on ( 0 , a 1 ] and decreasing on [ a 1 , 1 / 2 ] . Note that

f 4 ′ ( a ) = ∑ n = 1 ∞ n B 4 ( n ) a n − 1 , f 4 ′ ( 0 + ) = f 4 ′ ( 1 / 2 ) = 0 ,

and f 4 ′ ( a ) ≥ 0 on ( 0 , 1 / 2 ] . Therefore, we obtain the desired monotonicity and concavity of f 4 .

(3) Differentiating

(3.55) F 4 , n ( a ) = f 4 ( a ) − P 4 , n ( a ) ( 1 − 2 a ) 2 ( n + 1 ) = ∑ k = n + 1 ∞ A 4 ( 2 k ) ( 1 − 2 a ) 2 ( k − n − 1 ) = ∑ k = 0 ∞ A 4 ( 2 ( k + n + 1 ) ) ( 1 − 2 a ) 2 k

yields

(3.56) F 4 , n ( l ) ( a ) = ∑ k = [ ( l + 1 ) / 2 ] ∞ ( − 1 ) l 2 k ( 2 k ) ! ( 2 k − l ) ! A 4 ( 2 ( k + n + 1 ) ) ( 1 − 2 a ) 2 k − l ,

which leads to

F 4 , n ′ ( a ) = − 2 ∑ k = 1 ∞ 2 k A 4 ( 2 ( k + n + 1 ) ) ( 1 − 2 a ) 2 k − 1

and

F 4 , n ″ ( a ) = 4 ∑ k = 1 ∞ 2 k ( 2 k − 1 ) A 4 ( 2 ( k + n + 1 ) ) ( 1 − 2 a ) 2 k − 2 .

It follows from Lemma 2.1(1) that A 4 ( 2 n ) > 0 for all n ≥ 2 and n ∈ N , F 4 , n ′ ( a ) < 0 , F 4 , n ″ ( a ) > 0 . Therefore, we obtain the desired monotonicity and concavity of F 4 , n , and the desired inequality follows from the monotonicity of F 4 , n and F 4 , n ( 0 + ) = 1 − P 4 , n ( 0 ) , F 4 , n ( 1 / 2 + ) = A 4 ( 2 n + 2 ) .

(4) Let G 4 , n ( a ) = h 9 ( a ) / h 10 ( a ) , where h 10 ( a ) = a n + 1 and

(3.57) h 9 ( a ) = f 4 ( a ) − R 4 , n ( a ) .

Then h 9 ( m ) ( 0 + ) = h 10 ( m ) ( 0 + ) = 0 for all m ∈ N ∪ { 0 } with 0 ≤ m ≤ n and

(3.58) h 9 ( n + 1 ) ( a ) h 10 ( n + 1 ) ( a ) = f 4 ( n + 1 ) ( a ) ( n + 1 ) ! .

From Lemma 2.3, we know that G 4 , 2 n ( G 4 , ( 2 n − 1 ) ) have the same monotonicity with the function f 4 ( 2 n + 1 ) ( f 4 ( 2 n ) ) . Therefore, G 4 , 2 n ( G 4 , ( 2 n − 1 ) ) is increasing (decreasing) on ( 0 , 1 / 2 ] follows from the monotonicity of f 4 ( n ) .

Differentiating G 4 , n gives

( G 4 , n ( a ) ) ′ = h 9 ( a ) h 10 ( a ) ′ = a ( f 4 ( a ) − R 4 n ( a ) ) ′ − ( n + 1 ) ( f 4 ( a ) − R 4 , n ( a ) ) a n + 2 .

Let h 12 ( a ) = a n + 2 and

h 11 ( a ) = a [ f 2 ( a ) − R 2 , n ( a ) ] ′ − ( n + 1 ) [ f 2 ( a ) − R 2 , n ( a ) ] .

Then from (3.52), we clearly see that h 11 ( a ) can be rewritten as follows:

h 11 ( a ) = ∑ k = n + 1 ∞ ( k − n − 1 ) B 4 ( k ) a k .

Therefore, h 11 ( m ) ( 0 + ) = h 12 ( m ) ( 0 + ) = 0 for all m ∈ N ∪ { 0 } with 0 ≤ m ≤ n . Note that

(3.59) h 11 ( n + 1 ) ( a ) h 12 ( n + 1 ) ( a ) = f 4 ( n + 2 ) ( a ) ( n + 2 ) ! .

It follows from Lemma 2.3 that G 4 , n ′ has the same monotonicity with the function f 4 ( n + 2 ) . Therefore, we get the desired monotonicity of G 4 , 2 n ′ and G 4 , ( 2 n − 1 ) ′ , and the desired concavity (convexity) of G 4 , 2 n ′ ( G 4 , ( 2 n − 1 ) ′ ).

The double inequality (3.49) (3.50) follows easily from the monotonicity and concavity (convexity) of G 4 , 2 n ( G 4 , ( 2 n + 1 ) ) .□

Let n = 1 . Then Theorems 3.1, 3.3, 3.6 and 3.7 lead to Corollary 3.8 immediately.

Corollary 3.8

The inequalities

log 2 − A 1 ( 2 ) ( 1 − 2 a ) 2 − A 1 ( 4 ) ( 1 − 2 a ) 4 − ( λ ( 5 ) − 1 ) ( 1 − 2 a ) 5 ≤ b R ( a ) ≤ log 2 − A 1 ( 2 ) ( 1 − 2 a ) 2 − A 1 ( 4 ) ( 1 − 2 a ) 4 ,

(3.60) 1 − a + ( 4 log 2 − 2 ) a 3 ≤ b R ( a ) ≤ 1 − a + 2 ζ ( 3 ) a 3 − 4 ( 1 − 2 log 2 + ζ ( 3 ) ) a 4 ,

(3.61) 1 − a + 2 ζ ( 3 ) a 3 − 2 ζ ( 3 ) a 4 + 4 [ 12 + 8 log 2 − ζ ( 3 ) ] a 5 ≤ b R ( a ) ≤ 1 − a + 2 ζ ( 3 ) a 3 + 4 [ 12 + 8 log 2 − ζ ( 3 ) ] a 4 ,

9 log 2 4 + A 2 ( 2 ) [ ( 1 − 2 a ) 2 + ( 1 − 2 a ) 4 ] ≤ b ( 2 + b ) R ( a ) ≤ 9 log 2 4 + A 2 ( 2 ) [ ( 1 − 2 a ) 2 + ( 1 − 2 a ) 4 ] + [ ( 8 − 9 log 2 ) / 4 − 2 A 2 ( 2 ) ] ( 1 − 2 a ) 5 ,

(3.62) 2 − a − 2 a 2 + ( 4 ζ ( 3 ) + 1 ) a 3 + 2 [ 18 log 2 − 4 ζ ( 3 ) − 9 ] a 4 ≤ b ( 2 + b ) R ( a ) ≤ 2 − a − 2 a 2 + 2 ( 9 log 2 − 4 ) a 3 ,

(3.63) 2 − a − 2 a 2 + ( 9 log 2 − 6 ) a 4 ≤ b ( 2 + b ) R ( a ) ≤ 2 − a − 2 a 2 − 2 a 4 + 2 ( 9 log 2 − 4 ) a 5 ,

7 log 2 − 4 + ( 7 λ ( 3 ) + log 2 − 4 ) [ ( 1 − 2 a ) 2 + ( 1 − 2 a ) 4 ] < ( 2 − b ) R − 1 / b ,

(3.64) 1 a − 2 < ( 2 − b ) R − 1 / b ≤ 1 a − 2 + [ 56 log 2 − 8 ζ ( 3 ) − 30 ] a ,

(3.65) ( 2 − b ) R − 1 b < 1 a + 2 + min { ( 4 ζ ( 3 ) − 1 ) a , ( 28 log 2 − 16 ) a } ,

(3.66) A 4 ( 0 ) + A 4 ( 2 ) ( 1 − 2 a ) 2 + A 4 ( 4 ) ( 1 − 2 a ) 4 ≤ R ( a ) + b − 1 b ≤ A 4 ( 0 ) + A 4 ( 2 ) ( 1 − 2 a ) 2 + A 4 ( 4 ) ( 1 − 2 a ) 4 + ( F 4 , 1 ( 0 + ) − A 4 ( 4 ) ) ( 1 − 2 a ) 5 , − 1 + B 4 ( 2 ) a 2 − a 3 + 2 ( G 4 , 2 ( 1 / 2 ) + 1 ) a 4 ≤ R ( a ) + b − 1 b ≤ − 1 + B 4 ( 2 ) a 2 + G 4 , 2 ( 1 / 2 ) a 3

and

(3.67) − 1 + B 4 ( 2 ) a 2 + G 4 , 1 ( 1 / 2 ) a 4 ≤ R ( a ) + b − 1 b ≤ − 1 + B 4 ( 2 ) a 2 + B 4 ( 2 ) a 4 + 2 ( G 4 , 1 ( 1 / 2 ) − B 4 ( 2 ) ) a 5

hold for all a ∈ ( 0 , 1 / 2 ] , with equality in each instance if and only if a = 1 / 2 , where

A 1 ( 2 ) = log 2 − λ ( 3 ) = − 0.358653 … , A 1 ( 4 ) = λ ( 3 ) − λ ( 5 ) = 0.047276 … ,

A 2 ( 2 ) = ( 9 λ ( 3 ) − 10 log 2 ) / 4 = 0.633682 … ,

A 4 ( 0 ) = 4 log 2 − 15 4 = − 0.9774 … , A 4 ( 2 ) = 4 λ ( 3 ) − 17 4 = − 0.0426 … ,

A 4 ( 4 ) = 4 ( λ ( 3 ) − 1 ) = 0.2072 … , F 4 , 1 ( 0 + ) = − 1 − A 4 ( 0 ) − A 4 ( 2 ) = − 2.02 … .

G 4 , 1 ( 1 / 2 ) = 16 log 2 − 11 = − 3.90964512 …

G 4 , 2 ( 1 / 2 ) = 32 log 2 − 4 ζ ( 3 ) − 18 = − 0.0264894 … ,

and

B 4 ( 2 ) = 2 ζ ( 3 ) − 2 = 0.1035996 … .

Remark 3.9

It is not very difficult to prove that the upper and lower bounds for b R ( a ) in (3.60) are better than that in (3.61), the upper and lower bounds for b ( 2 + b ) R ( a ) in (3.62) are better than that in (3.63), the upper bound for ( 2 − b ) R ( a ) − 1 / b in (3.65) is better than that in (3.64), and the upper and lower bounds for R ( a ) + b − 1 / b in (3.66) are better than that in (3.67). We leave the details of the proof to the interested readers.

4 Conclusion

In the article, we have established some monotonicity and convexity (concavity) properties for certain combinations of polynomials and the Ramanujan R -function, R ( a ) = − 2 γ − ψ ( a ) − ψ ( 1 − a ) by use of the L’hôpital’s monotone rule, and presented serval novel bounds for the Ramanujan R -function, which are the refinements and improvements of the results given in [54]. The proposed method and technology may lead to a lot of follow-up work in the future.

Acknowledgements

The authors express their gratitude to the referees for very helpful and detailed comments and suggestions, which have significantly improved the presentation of this paper.

  1. Funding information: The research was supported by the Natural Science Foundation of China (Grant Nos.: 11401531, 61673169, 11701176, 11626101, and 11601485).

  2. Author contributions: All authors contributed equally to the manuscript, and they read and approved the final manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2021-03-11
Revised: 2022-03-07
Accepted: 2022-04-19
Published Online: 2022-09-01

© 2022 Chuan-Yu Cai et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0045
Schlagwörter für diesen Artikel
Ramanujan R-function; Gaussian hypergeometric function; Riemann zeta function; generalized elliptic integral; monotonicity; convexity
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Artikel in diesem Heft

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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