Home Mathematics Approximations related to the complete p-elliptic integrals
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Approximations related to the complete p-elliptic integrals

  • Genhong Zhong , Xiaoyan Ma EMAIL logo and Fei Wang
Published/Copyright: September 26, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

In this paper, the authors present some monotonicity properties for certain functions involving the complete p-elliptic integrals of the first and second kinds, by showing the monotonicity and concavity-convexity properties of certain combinations defined in terms of K p , E p and the inverse hyperbolic tangent arth p , which is of importance in the computation of the generalized pi and in the elementary proof of Ramanujan’s cubic transformation. By these results, several well-known results for the classical complete elliptic integrals including its bounds and logarithmic inequalities are remarkably improved.

Keywords: complete p-elliptic integrals; monotonicity; inequality; bounds inequalities; logarithmic inequalities
MSC 2010: 33C75; 33E05; 33F05

1 Introduction

Throughout this paper, we denote the set of positive integers (the inverse hyperbolic tangent) by N ( arth , respectively), and let r ′ p + r p = 1 for each r ∈ [ 0 , 1 ] and p ∈ ( 1 , ∞ ) . For r ∈ ( 0 , 1 ) , the complete elliptic integrals of the first and second kinds are defined as follows:

(1.1) K = K ( r ) = ∫ 0 π / 2 ( 1 − r 2 sin 2 t ) − 1 / 2 d t , K ′ = K ′ ( r ) = K ( r ′ ) , K ( 0 ) = π / 2 , K ( 1 ) = ∞ , and

(1.2) E = E ( r ) = ∫ 0 π / 2 ( 1 − r 2 sin 2 t ) 1 / 2 d t , E ′ = E ′ ( r ) = E ( r ′ ) , E ( 0 ) = π / 2 , E ( 1 ) = 1 ,

respectively, which are the particular cases of the Gaussian hypergeometric function:

(1.3) F ( a , b , c , x ) = F 1 2 ( a , b ; c ; x ) = ∑ n = 0 ∞ ( a ) n ( b ) n ( c ) n n ! x n ( ∣ x ∣ < 1 )

for a , b , c ∈ C with c ≠ 0 , − 1 , − 2 , … , where ( a ) n = a ( a + 1 ) ( a + 2 ) ⋯ ( a + n − 1 ) for n ∈ N , and ( a ) 0 = 1 for a ≠ 0 . As a matter of fact, it is well known that

(1.4) K ( r ) = π 2 F 1 2 , 1 2 ; 1 ; r 2 and E ( r ) = π 2 F − 1 2 , 1 2 ; 1 ; r 2

[1, 17.3.9–17.3.10]. For the properties of the complete elliptic integrals, the readers can refer to [2,3,4, 5,6,7, 8,9,10, 11,12,13, 14,15,16, 17,18,19, 20,21] and the bibliographies therein.

As far as the complete elliptic integrals of the first and second kinds are concerned, there are kinds of bounds for them in terms of the inverse hyperbolic tangent function [1, 15.1.4], that is,

(1.5) arth r r = F 1 2 , 1 ; 3 2 ; r 2 = ∑ n = 0 ∞ 1 2 n + 1 r 2 n .

In 1992, Anderson et al. mentioned such bound, and they presented the double inequality [7, Theorem 3.10]

(1.6) arth r r 1 / 2 < 2 π K ( r ) < arth r r

for r ∈ ( 0 , 1 ) . This was improved by Alzer and Qiu in [4, Theorem 18] as

(1.7) arth r r α 1 < 2 π K ( r ) < arth r r β 1

for r ∈ ( 0 , 1 ) with the best possible constants α 1 = 3 / 4 and β 1 = 1 . In [15, Theorem 1.2], Wang et al. obtained the double inequality

(1.8) π 2 − 3 π 16 r − ( 1 − r 2 ) arth r r < E ( r ) < π 2 − π 2 − 1 r − ( 1 − r 2 ) arth r r

holds for all r ∈ ( 0 , 1 ) .

In 2014, Takeuchi [22] introduced a form of the generalized complete elliptic integrals as an application of generalized trigonometric functions. The complete p-elliptic integrals of the first and second kinds are, respectively, defined as follows: for p ∈ ( 1 , ∞ ) and r ∈ [ 0 , 1 ) ,

(1.9) K p = K p ( r ) = ∫ 0 π p / 2 d θ ( 1 − r p sin p p θ ) 1 − 1 / p = π p 2 F 1 p , 1 − 1 p ; 1 ; r p , K p ( 0 ) = π p 2 , K p ( 1 − ) = ∞

and

(1.10) E p = E p ( r ) = ∫ 0 π p / 2 ( 1 − r p sin p p θ ) 1 / p d θ = π p 2 F 1 p , − 1 p ; 1 ; r p , E p ( 0 ) = π p 2 , E p ( 1 − ) = 1 ,

where sin p θ is the generalized sine function, defined by the inverse function of

arcsin p ( θ ) = ∫ 0 θ 1 ( 1 − t p ) 1 / p d t , 0 ≤ θ ≤ 1 , π p 2 = arcsin p ( 1 ) = ∫ 0 1 1 ( 1 − t p ) 1 / p d t = π / p sin ( π / p ) .

In [23], it was obtained that, for p ∈ ( 1 , ∞ ) ,

(1.11) arth p r r < K p ( r ) < π p 2 arth p r r ,

where

(1.12) arth p r r = F 1 p , 1 ; 1 p + 1 ; r p = ∑ n = 0 ∞ 1 p n + 1 r p n

is the generalized inverse hyperbolic tangent function.

Recently, Wang and Qi [24, Theorem 1.3] obtained the bounds

(1.13) π p , q 2 arth q r r ( 1 − α 3 r q ) < K p , q ( r ) < π p , q 2 arth q r r ( 1 − β 3 r q )

for r ∈ ( 0 , 1 ) , p , q ∈ ( 1 , ∞ ) with the best weights α 3 = 1 − 2 / π p , q and β 3 = 1 / [ p q ( q + 1 ) ] , where

π p , q = 2 ∫ 0 1 1 ( 1 − t q ) 1 / p d t .

When p = q , K p , q ( r ) = K ( r ) , arth q ( r ) = arth p r , π p , q = π p . And as we know, for p = 2 , these functions reduce to well-known special cases K p ( r ) = K ( r ) , E p ( r ) = E ( r ) , arth p ( r ) = arth r , π p = π , and numerous properties of the complete p-elliptic integrals have been obtained (cf. [22,23,24, 25,26,27, 28,29]) and bibliographies therein.

Inspired by the inequalities (1.7), (1.8), (1.11), and (1.13), the topic of studying the properties of the complete p-elliptic integrals is to approximate K p ( r ) and E p ( r ) by means of the function r ↦ ( arth p r ) / r .

The main purpose of this paper is to present several new monotonicity properties of the complete p-elliptic integrals. By obtained results, some approximations of K p ( r ) and E p ( r ) by certain combinations in terms of arth p ( r ) / r and polynomials are derived.

2 Preliminaries and proofs

In the sequel, we sometimes omit the variable r of the generalized and complete elliptic integrals when there is no confusion, and always let arth p denote the generalized inverse hyperbolic tangent.

In order to prove our main results stated in Section 3, let us recall the following well-known formulas [22, Proposition 2.1]: For r ∈ ( 0 , 1 ) , p ∈ ( 1 , ∞ ) ,

(2.1) d K p d r = E p ( r ) − r ′ p K p ( r ) r r ′ p , d E p d r = E p ( r ) − K p ( r ) r ,

(2.2) d d r ( K p − E p ) = r p − 1 E p ( r ) r ′ p ,

(2.3) d arth p d r = 1 1 − r p = 1 r ′ p ,

which will be frequently applied later.

Now we present several lemmas in this section.

Lemma 2.1

(See [8, Theorem 1.25]) Let − ∞ < a < b < ∞ , f , g : [ a , b ] → R be continuous on [ a , b ] and be differentiable on ( a , b ) such that g ′ ( x ) ≠ 0 on ( a , b ) . Then both the functions [ f ( x ) − f ( a ) ] / [ g ( x ) − g ( a ) ] and [ f ( x ) − f ( b ) ] / [ g ( x ) − g ( b ) ] are (strictly) increasing (decreasing) on ( a , b ) if f ′ ( x ) / g ′ ( x ) is (strictly) increasing (decreasing) on ( a , b ) .

Lemma 2.2

(See [16, Theorem 2.1]) Suppose that the power series f ( x ) = ∑ n = 0 ∞ a n x n and g ( x ) = ∑ n = 0 ∞ b n x n have the radius of convergence r > 0 with b n > 0 for all n ∈ { 0 , 1 , 2 , … } . Let h ( x ) = f ( x ) / g ( x ) and H f , g = ( f ′ / g ′ ) g − f , then the following statements are true:

  1. If the nonconstant sequence { a n / b n } ∣ n = 0 ∞ is increasing (decreasing), then h ( x ) is strictly increasing (decreasing) on ( 0 , r ) ;

  2. If the nonconstant sequence { a n / b n } is increasing (decreasing) for 0 < n ≤ n 0 and decreasing (increasing) for n > n 0 , then the function h is strictly increasing (decreasing) on ( 0 , r ) if and only if H f , g ( r − ) ≥ ( ≤ ) 0 . While if H f , g ( r − ) < ( > ) 0 , then exists δ ∈ ( 0 , r ) such that h ( x ) is strictly increasing (decreasing) on ( 0 , δ ) and strictly decreasing (increasing) on ( δ , r ) .

Our first lemma presents several properties of the generalized inverse hyperbolic tangent function arth p .

Lemma 2.3

  1. For p > 1 , the function g 1 ( r ) ≡ [ r − r ′ p arth p r ] / r p + 1 is strictly increasing and convex from ( 0 , 1 ) onto ( p / ( p + 1 ) , 1 ) .

  2. For p > ( 3 + 17 ) / 4 , the function g 2 ( r ) ≡ [ 1 + ( p − 2 ) r p ] g 1 ( r ) is increasing and convex from ( 0 , 1 ) onto ( p / ( p + 1 ) , p − 1 ) .

  3. For p ∈ ( ( 3 + 17 ) / 4 , 1 + 2 ) , the function g 3 ( r ) ≡ [ r + ( p − 1 ) r ′ p arth p r ] / ( r r ′ ) is strictly increasing and convex from ( 0 , 1 ) onto ( p , ∞ ) .

  4. For p ∈ ( ( 3 + 17 ) / 4 , 1 + 2 ) , the function g 4 ( r ) ≡ ( r − r ′ p arth p r ) / [ r ′ ( arth p r − r ) ] is strictly increasing from ( 0 , 1 ) onto ( p , ∞ ) .

Proof

  1. By (1.12), g 1 ( r ) can be written as follows:

    (2.4) g 1 ( r ) = 1 − r ′ p ( arth p r / r ) r p = p ∑ n = 0 ∞ r p n ( p n + 1 ) ( p n + p + 1 ) ,

    yielding the monotonicity and convexity for g 1 and the limiting value g 1 ( 0 + ) = p / ( p + 1 ) . Clearly, g 1 ( 1 − ) = 1 .

  2. Clearly, g 2 ( 0 ) = p / ( p + 1 ) , g 2 ( 1 ) = p − 1 . By (2.4), g 2 can be written as follows:

    (2.5) g 2 ( r ) = p ∑ n = 0 ∞ r p n ( p n + 1 ) ( p n + p + 1 ) + ( p − 2 ) ∑ n = 0 ∞ r p ( n + 1 ) ( p n + 1 ) ( p n + p + 1 ) = p p + 1 + p ∑ n = 0 ∞ p ( p − 1 ) n + ( 2 p 2 − 3 p − 1 ) ( p n + 1 ) ( p n + p + 1 ) ( p n + 2 p + n ) r p ( n + 1 ) = p p + 1 + p ∑ n = 0 ∞ f ( n , p ) ( p n + 1 ) ( p n + p + 1 ) ( p n + 2 p + n ) r p ( n + 1 ) ,

    where f ( n , p ) = p ( p − 1 ) n + ( 2 p 2 − 3 p − 1 ) = ( n + 2 ) p 2 − ( n + 3 ) p − 1 . It is easy to see that f ( n , p ) is strictly increasing with respect to p . Then f ( n , p ) > f ( n , ( 3 + 17 ) / 4 ) = [ ( 7 + 17 ) n ] / 8 ≥ 0 . Hence, the monotonicity and convexity properties of g 2 follow from (2.5).

  3. The limiting values of g 3 are clear. By (2.3) and by differentiation, we obtain

    (2.6) g 3 ′ ( r ) = [ p − p ( p − 1 ) r p − 1 arth p r ] r r ′ − [ r + ( p − 1 ) r ′ p arth p r ] ( 1 − 2 r p ) / r ′ ( p − 1 ) ( r r ′ ) 2 = r [ p r ′ p − 1 + 2 r p ] − ( p − 1 ) r ′ p arth p r [ ( p − 2 ) r p + 1 ] r 2 r ′ ( p + 1 ) = ( p − 1 ) ( r − r ′ p arth p r ) + ( 2 − p ) r p [ r + ( p − 1 ) r ′ p arth p r ] r 2 r ′ ( p + 1 ) = r p − 1 r ′ ( p + 1 ) ⋅ r − r ′ p arth p r r p + 1 ( p − 1 ) + ( 2 − p ) r p 1 + p r ′ p arth p r r − r ′ p arth p r

    = r p − 1 r ′ ( p + 1 ) ⋅ r − r ′ p arth p r r p + 1 ( p − 1 ) + ( 2 − p ) r p 1 − p + p r r − r ′ p arth p r = r p − 1 r ′ ( p + 1 ) ⋅ r − r ′ p arth p r r p + 1 ⋅ ( p − 1 ) + ( 2 − p ) ( 1 − p ) r p + p r p + 1 r − r ′ p arth p r = r p − 1 r ′ ( p + 1 ) ⋅ { ( p − 1 ) [ 1 + ( p − 2 ) r p ] g 1 ( r ) + p ( 2 − p ) } = r p − 1 r ′ ( p + 1 ) ⋅ [ ( p − 1 ) g 2 ( r ) + p ( 2 − p ) ] = r p − 1 r ′ ( p + 1 ) ⋅ G 1 ( r ) ,

    where G 1 ( r ) = ( p − 1 ) g 2 ( r ) + p ( 2 − p ) . Hence, by the part (2), G 1 is increasing on (0,1) for p > ( 3 + 17 ) / 4 . On the other hand, it is easy to verify that

    G 1 ( 0 ) = p ( − p 2 + 2 p + 1 ) p + 1 > 0

    for p ∈ ( ( 3 + 17 ) / 4 , 1 + 2 ) . Consequently, by (2.6), g 3 ′ is strictly increasing on ( 0 , 1 ) with g 3 ′ ( 0 ) = 0 , and hence, the monotonicity and convexity properties of g 3 follow from part (2).

  4. Let G 2 ( r ) = r / r ′ − r ′ ( p − 1 ) arth p r and G 3 ( r ) = arth p r − r . Then G 2 ( 0 ) = G 3 ( 0 ) = 0 , g 4 ( r ) = G 2 ( r ) / G 3 ( r ) and

    (2.7) G 2 ′ ( r ) / G 3 ′ ( r ) = g 3 ( r ) .

Hence, the monotonicity of g 4 follows from Lemma 2.1 and part (3). Applying l’Hôpital’s rule and (2.7), g 4 ( 0 + ) = p and g 4 ( 1 − ) = ∞ .□

Lemma 2.4

Given p ∈ [ 2 , ∞ ) , the function h ( r ) ≡ E p / [ r ′ ( p − 1 ) K p ] + ( p − 2 ) r ′ is strictly increasing and convex from ( 0 , 1 ) onto ( p − 1 , ∞ ) .

Proof

Clearly, h ( 0 ) = p − 1 and h ( 1 ) = ∞ . Differentiation gives

(2.8) h ′ ( r ) = r ′ ( p − 1 ) ( E p − K p ) K p r − E p E p − r ′ p K p r r ′ − ( p − 1 ) r p − 1 r ′ K p ⋅ [ r ′ ( p − 1 ) K p ] − 2 − ( p − 2 ) r r ′ p − 1 = r r ′ p − 1 ⋅ E p r ′ p K p ( p − 1 ) − ( K p − E p ) r ′ p r p E p + E p − r ′ p K p r p K p + ( p − 2 ) r ′ p K p E p = r r ′ p − 1 ⋅ E p r ′ p K p [ ( p − 1 ) − H 1 ( r ) ] ,

where

H 1 ( r ) = ( K p − E p ) r ′ p r p E p + E p − r ′ p K p r p K p + ( p − 2 ) r ′ p K p E p .

By [23, Lemma 3.4(2),(3),(5)], for p ≥ 2 , we see that H 1 ( r ) is decreasing from (0,1) onto ( 0 , p − 1 ) . Hence, it follows from (2.8) that h ′ is the product of three positive and increasing functions. This yields the monotonicity and convexity properties of h .□

Lemma 2.5

The function j ( x ) = 1 / sin x − 1 / x − 1 / π is increasing from ( 0 , π ) onto ( − 1 / π , ∞ ) , and exists x 0 = 1.451765874910260 ⋯ ∈ ( 0 , π ) for j ( x 0 ) = 0 .

Proof

By the Taylor expansion of x / sin x ([30, Equality (2.15)]), we obtain

(2.9) j ( x ) = ∑ k = 1 ∞ 2 2 k − 2 ( 2 k ) ! ∣ B 2 k ∣ x 2 k − 1 − 1 / π = 1 6 x + ∑ k = 2 ∞ 2 2 k − 2 ( 2 k ) ! ∣ B 2 k ∣ x 2 k − 1 − 1 / π ,

where B n ( n ∈ N 0 , N 0 = N ∪ { 0 } ) are the Bernoulli numbers, defined by

t e t − 1 = ∑ n = 0 ∞ B n t n n ! , ∣ t ∣ < 2 π .

By differentiation,

j ′ ( x ) = 1 6 + ∑ k = 2 ∞ 2 2 k − 2 ( 2 k ) ! ∣ B 2 k ∣ ( 2 k − 1 ) x 2 k − 2 .

It is easy to see that for all x ∈ ( 0 , π ) , j ′ ( x ) > 0 . Consequently, j ( x ) is increasing on ( 0 , π ) . It follows from (2.9) that the limits j ( 0 ) = − 1 / π and j ( π ) = ∞ . By the mathematical software Maple, we compute that j ( x 0 ) = 0 .□

3 Main results

In this section, we reveal some monotonicity properties of K p ( r ) and E p ( r ) , and some of which remarkably improve the related well-known results for them such as [4, Theorem 18] and [15, Theorem 1.2].

Theorem 3.1

  1. For r ∈ ( 0 , 1 ) , p ∈ [ 2 , 1 + 2 ] , the function

    f 1 ( r ) ≡ log ( 2 K p ( r ) / π p ) log ( arth p r / r )

    is strictly increasing from ( 0 , 1 ) onto ( 1 − 1 / p 2 , 1 ) . In particular, for all r ∈ ( 0 , 1 ) , p ∈ [ 2 , 1 + 2 ] , the double inequality

    (3.1) π p 2 arth p r r α 1 < K p ( r ) < π p 2 arth p r r β 1

    holds with the best possible constants α 1 = 1 − 1 / p 2 and β 1 = 1 .

  2. For r ∈ ( 0 , 1 ) , p ∈ ( 1 , ∞ ) , the function

    f 2 ( r ) = r ( 2 K p ( r ) / π p − 1 ) arth p r − r

    is strictly decreasing from ( 0 , 1 ) onto ( 2 / π p , 1 − 1 / p 2 ) . In particular, for all r ∈ ( 0 , 1 ) , p ∈ ( 1 , ∞ ) , the double inequality

    (3.2) 1 − α 2 + α 2 arth p r r < 2 π p K p < 1 − β 2 + β 2 arth p r r

    holds with the best possible constants α 2 = 2 / π p and β 2 = 1 − 1 / p 2 .

Proof

(1) Let

F 1 ( r ) = log 2 K p π p , F 2 ( r ) = log arth p r r , F 3 ( r ) = K p − arth p r r E p , F 4 ( r ) = 1 − r ′ p arth p r r K p .

Then F 1 ( 0 + ) = F 2 ( 0 + ) = F 3 ( 0 + ) = F 4 ( 0 + ) = 0 , f 1 ( r ) = F 1 ( r ) / F 2 ( r ) , and by differentiation,

(3.3) F 1 ′ ( r ) F 2 ′ ( r ) = 1 − r K p − E p arth p r ( r − r ′ p arth p r ) K p = 1 − F 3 ( r ) F 4 ( r ) ,

(3.4) F 3 ′ ( r ) F 4 ′ ( r ) = E p − r ′ p K p r r ′ p + K p − E p r 2 arth p r − E p r − r ′ p arth p r r 2 r ′ p × ( p r p + 2 r ′ p ) arth p r − 2 r r 2 K p + ( r − r ′ p arth p r ) E p r 2 r ′ p − 1 = [ ( arth p r − r ) r ′ p K p ] { ( r − r ′ p arth p r ) E p + [ ( p r p + 2 r ′ p ) arth p r − 2 r ] r ′ p K p } − 1 = p + r − r ′ p arth p r arth p r − r ⋅ E p r ′ p K p + ( p r p + 2 r ′ p − p ) arth p r + ( p − 2 ) r arth p r − r − 1 = p + r − r ′ p arth p r arth p r − r ⋅ E p r ′ p K p + ( 2 − p ) r ′ p arth p r + ( p − 2 ) r arth p r − r − 1 = [ p + g 4 ( r ) h ( r ) ] − 1 ,

where g 4 and h are, respectively, defined as in Lemmas 2.3(4) and in 2.4. It follows from (3.4), Lemmas 2.3(4) and 2.4 that the function F 3 ′ / F 4 ′ is strictly decreasing on (0,1), and is F 3 / F 4 by Lemma 2.1. Consequently, it follows from (3.3) and Lemma 2.1 that f 1 is strictly increasing on (0,1).

By Lemma 2.1 (l’Hôpital’s rule), Lemma 2.3(4), Lemma 2.4, and by (3.3)–(3.4), we obtain the limiting values:

f 1 ( 0 + ) = 1 − lim r → 0 + 1 p + g 4 ( r ) h ( r ) = 1 − 1 p 2 and f 1 ( 1 − ) = 1 − lim r → 1 − 1 p + g 4 ( r ) h ( r ) = 1 .

By the monotonicity of f 1 , the double inequality (3.1) holds with α 1 = 1 − 1 / p 2 and β 1 = 1 .

It is easy to see that for all r ∈ ( 0 , 1 ) and p ∈ [ 2 , 1 + 2 ] ,

K p ( r ) > π p 2 arth p r r α 1 ⇔ α 1 ≤ inf r ∈ ( 0 , 1 ) f 1 ( r ) = f 1 ( 0 + ) = 1 − 1 / p 2

and

K p ( r ) < π p 2 arth p r r β 1 ⇔ β 1 ≥ sup r ∈ ( 0 , 1 ) f 1 ( r ) = f 1 ( 1 − ) = 1 .

Hence, the last assertion follows.

(2) Let F 5 ( r ) = 2 K p ( r ) / π p − 1 , F 6 ( r ) = arth p r / r − 1 . By using (1.9) and (1.12), we obtain

(3.5) f 2 ( r ) = F 5 ( r ) F 6 ( r ) = ∑ n = 1 ∞ 1 p , n 1 − 1 p , n ( n ! ) 2 r n p ∑ n = 1 ∞ 1 p n + 1 r n p = ∑ n = 1 ∞ a n r n p ∑ n = 1 ∞ b n r n p ,

where

a n = 1 p , n 1 − 1 p , n ( n ! ) 2 , and b n = 1 p n + 1 .

Let c n = a n / b n , then

c n + 1 c n = 1 − 1 p 2 ( n + 1 ) 2 < 1 .

We conclude from (3.5) that f 2 is strictly decreasing on (0,1).

It easy to see that F 5 ( 0 ) = 0 and F 6 ( 0 + ) = 0 . Applying (2.1), we have

F 5 ′ ( r ) F 6 ′ ( r ) = 2 π p r ( E p − r ′ p K p ) r − r ′ p arth p r = 2 π p ( E p − r ′ p K p ) / r p ( r − r ′ p arth p r ) / r p + 1 = 2 π p F 7 ( r ) g 1 ( r ) ,

where F 7 ( r ) = [ E p − r ′ p K p ] / r p , g 1 ( r ) is in Lemma 2.3(1). By l’Hôpital’s rule, Lemma 2.3(1), and [23, Lemma 3.4 (3)], we obtain the limiting values

f 2 ( 0 + ) = lim r → 0 + F 5 ′ ( r ) F 6 ′ ( r ) = 2 π p lim r → 0 + F 7 ( r ) g 1 ( r ) = 1 − 1 p 2 and f 2 ( 1 − ) = lim r → 1 − F 5 ′ ( r ) F 6 ′ ( r ) = 2 π p lim r → 1 − F 7 ( r ) g 1 ( r ) = 2 π p . □

Theorem 3.2

For p ∈ ( 1 , ∞ ) , let

p 0 = 1 6 ( 135 + 6 249 ) 1 / 3 + 7 2 ( 135 + 6 249 ) 1 / 3 + 1 2 ≈ 2.092193586 ⋯

be the unique solution to the equation 2 p 3 − 3 p 2 − 2 p − 1 = 0 and p ∗ = 2.16398023116776 ⋯ be the unique solution to the equation 1 / p + 1 − π / [ p sin ( π / p ) ] = 0 , and the function f 3 is defined on ( 0 , 1 ) by

f 3 ( r ) ≡ r ( π p / 2 − E p ( r ) ) r − r ′ p arth p r .

Then the following statements are true:

  1. If p ∈ ( 1 , p 0 ] , then f 3 is strictly decreasing from ( 0 , 1 ) onto ( π p / 2 − 1 , [ ( p + 1 ) π p ] / ( 2 p 3 ) ) . In particular, for all r ∈ ( 0 , 1 ) , p ∈ ( 1 , p 0 ] , the double inequality

    (3.6) π p 2 − α 3 + α 3 r ′ p arth p r r < E p ( r ) < π p 2 − β 3 + β 3 r ′ p arth p r r

    holds with the best possible constants α 3 = [ ( p + 1 ) π p ] / ( 2 p 3 ) and β 3 = π p / 2 − 1 .

  2. If p ∈ ( p 0 , p ∗ ) , then there exists unique r 0 ∈ ( 0 , 1 ) such that f 3 ( r ) is strictly increasing on ( 0 , r 0 ) , and strictly decreasing on ( r 0 , 1 ) . Consequently, for r ∈ ( 0 , 1 ) , one has

    (3.7) E p ( r ) < π p 2 − min π p 2 − 1 , ( p + 1 ) π p 2 p 3 1 − r ′ p arth p r r .

  3. If p ∈ ( p ∗ , ∞ ) , then f 3 is strictly increasing from (0,1) onto ( [ ( p + 1 ) / π p ] / ( 2 p 3 ) , π p / 2 − 1 ) , and the reverse inequality of (3.6) holds for all r ∈ ( 0 , 1 ) .

Proof

Clearly, f 3 ( 1 ) = π p / 2 − 1 . Let F 8 ( r ) = r [ π p / 2 − E p ( r ) ] , F 9 ( r ) = r − r ′ p arth p r . By using the series expansion (1.10) and (1.12), we obtain

(3.8) f 3 ( r ) = F 8 ( r ) F 9 ( r ) = π p / 2 − E p ( r ) 1 − r ′ p arth p r / r = − π p 2 ∑ n = 1 ∞ ( 1 / p , n ) ( − 1 / p , n ) ( n ! ) 2 r n p 1 − ∑ n = 0 ∞ 1 p n + 1 r n p + ∑ n = 0 ∞ 1 p n + 1 r ( 1 + n ) p = π p 2 p ∑ n = 1 ∞ ( 1 / p , n ) ( 1 − 1 / p , n ) ( n ! ) 2 ( p n − 1 ) r n p ∑ n = 1 ∞ 1 ( p n − p + 1 ) ( p n + 1 ) r n p = π p 2 p ∑ n = 1 ∞ A n r n p ∑ n = 1 ∞ B n r n p = π p 2 p ∑ n = 0 ∞ A n + 1 r n p ∑ n = 0 ∞ B n + 1 r n p ,

where

A n = ( 1 / p , n ) ( 1 − 1 / p , n ) ( n ! ) 2 ( p n − 1 ) , and B n = 1 ( p n − p + 1 ) ( p n + 1 ) .

By (3.8), we obtain

(3.9) f 3 ( 0 + ) = lim r → 0 + f 3 ( r ) = π p 2 p A 1 B 1 = ( p + 1 ) π p 2 p 3 .

Let C n = A n / B n , then

(3.10) C n + 1 C n − 1 = p ( p 2 − 2 p − 1 ) n + ( p 3 − p 2 − p − 1 ) p 2 ( n + 1 ) 2 ( p n − p + 1 ) = p J 1 ( p ) n + J 2 ( p ) p 2 ( n + 1 ) 2 ( p n − p + 1 ) = J ( p , n ) p 2 ( n + 1 ) 2 ( p n − p + 1 ) ,

where

J ( p , n ) = p ( p 2 − 2 p − 1 ) n + ( p 3 − p 2 − p − 1 ) , J 1 ( p ) = p 2 − 2 p − 1 , J 2 ( p ) = p 3 − p 2 − p − 1 .

Case 1. p ∈ ( 1 , p 0 ) .

p J 1 ( p ) = p ( p 2 − 2 p − 1 ) = p ( p + 2 − 1 ) [ p − ( 1 + 2 ) ] .

Clearly, p J 1 ( p ) < 0 for p < p 0 < 1 + 2 , we obtain

(3.11) J ( p , n ) < J ( p , 1 ) = p J 1 ( p ) + J 2 ( p ) = 2 p 3 − 3 p 2 − 2 p − 1 .

By differentiation,

J ′ ( p , 1 ) = 6 ( p 2 − p − 1 / 3 ) = 6 p − 3 + 21 6 p − 3 − 21 6 ,

then J ( p , 1 ) is strictly decreasing from ( 1 , ( 3 + 21 ) / 6 ) onto ( − 4.28211277 ⋯ , − 4 ) and strictly increasing from ( ( 3 + 21 ) / 6 , p 0 ) onto ( − 4.28211277 ⋯ , 0 ) , then J ( p , 1 ) < 0 . From (3.10) and (3.11), we obtain that C n + 1 / C n < 1 for n ≥ 1 , and thereby C n is strictly decreasing with respect to n . With an application of Lemma 2.2(1) and (3.8), the monotonicity of f 3 on ( 0 , 1 ) in this case is expressed as follows.

Case 2. p ∈ ( p 0 , ∞ ) .

By using (2.1) and (2.3), we obtain

(3.12) F 8 ′ ( r ) F 9 ′ ( r ) = ( π p / 2 − E p ) + ( K p − E p ) p r p − 1 arth p r ,

and thus,

(3.13) H F 8 , F 9 ( r ) = F 8 ′ F 9 ′ F 9 − F 8 = ( π p / 2 − E p ) + ( K p − E p ) p r p − 1 arth p r [ r − r ′ p arth p r ] − r π p 2 − E P = r 2 ( π p / 2 − 2 E p ) p arth p r + r p r K p arth p r ⋅ r − r ′ p arth p r r p + 1 − r π p 2 − E P = r 2 ( π p / 2 − 2 E p ) p arth p r + r p r K p arth p r ⋅ g 1 ( r ) + r E P − π p 2 .

It is not difficult to verify that

lim r → 1 r K p ( r ) arth p r = 1 ,

and thus

(3.14) H F 8 , F 9 ( 1 − ) = lim r → 1 − H F 8 , F 9 ( r ) = 1 p + 1 − π p 2 = 1 p + 1 − π p sin ( π / p ) = − π p ⋅ j π p ,

where j is in Lemma 2.5.

Then there exists unique zero point p ∗ = 2.16398023116776 ⋯ ∈ ( 1 , ∞ ) such that H F 8 , F 9 ( 1 − ) = 0 , H F 8 , F 9 ( 1 − ) < 0 for ( p 0 , p ∗ ) and H F 8 , F 9 ( 1 − ) > 0 for ( p ∗ , ∞ ) .

(i). p ∈ ( p 0 , p ∗ ) .

Then (3.10) implies that there exists n 0 > 1 such that sequence A n / B n is increasing for 1 ≤ n ≤ n 0 and decreasing for n > n 0 , for the limiting value of H F 8 , F 9 ( r ) at 1. It follows from (3.14) and Lemma 2.2(2) that H F 8 , F 9 ( 1 − ) < 0 for p ∈ ( p 0 , p ∗ ) , so f 3 ( r ) is piecewise monotone on (0,1), and therefore, the inequality (3.7) follows from

f 3 ( r ) > min π p 2 − 1 , ( p + 1 ) π p 2 p 3 .

(ii). p ∈ ( p ∗ , ∞ )

Applying Lemma 2.2 (2) and (3.14), f 3 ( r ) is strictly increasing from ( 0 , 1 ) onto ( [ ( p + 1 ) π p ] / ( 2 p 3 ) , π p / 2 − 1 ) if and only if p ≥ p ∗ , yielding the third assertion.□

Remark 3.3

  1. When p = 2 , the inequality (3.1) in Theorem 3.1 reduces to the inequality (1.7).

  2. By using (1.12), we clearly see that

    (3.15) arth p r r = 1 + ∑ n = 1 ∞ r p n p n + 1 > 1 ,

    (3.16) r ′ p arth p r r = ( 1 − r p ) ∑ n = 0 ∞ r p n p n + 1 = 1 + ∑ n = 1 ∞ r p n p n + 1 − ∑ n = 0 ∞ r p ( n + 1 ) p n + 1 = 1 − p ∑ n = 0 ∞ r p ( n + 1 ) ( p n + 1 ) ( p n + p + 1 ) < 1 .

    The inequality (3.15) implies that the upper bound for K p ( r ) given in (3.2) is better than that given in (1.11) for all r ∈ ( 0 , 1 ) . It is clear that 2 / π p ( = [ p sin ( π / p ) ] / π ) is increasing from ( 1 , ∞ ) onto ( 0 , 1 ) . This yields 1 − 2 / π p > 0 for p ∈ ( 1 , ∞ ) . Therefore, the lower bound for K p ( r ) given in (3.2) is better than that given in (1.13) for q = p by the inequality (3.16).

  3. When p = 2 , inequality (3.6) in Theorem 3.2 reduces to inequality (1.8).

Acknowledgements

The authors wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions.

  1. Funding information: This work was supported by the research project for Keyi College of Zhejiang Sci-Tech University (KY2021004), the National Natural Science Foundation of China (Grant No. 11671360), the Project for Combination of Education and Research Training at Zhejiang Institute of Mechanical and Electrical Engineering (Grant No. S027120206), and the project for technology innovation team of Zhejiang Institute of Mechanical and Electrical Engineering.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

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

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Received: 2021-03-25
Revised: 2022-03-26
Accepted: 2022-08-09
Published Online: 2022-09-26

© 2022 Genhong Zhong et al., published by De Gruyter

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

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https://doi.org/10.1515/math-2022-0493
Keywords for this article
complete p-elliptic integrals; monotonicity; inequality; bounds inequalities; logarithmic inequalities
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  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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