Home Inequalities for the generalized trigonometric and hyperbolic functions
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Inequalities for the generalized trigonometric and hyperbolic functions

  • Xiaoyan Ma EMAIL logo , Xiangbin Si , Genhong Zhong and Jianhui He
Published/Copyright: December 29, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

Abstract

In this paper, the authors present some inequalities of the generalized trigonometric and hyperbolic functions which occur in the solutions of some linear differential equations and physics. By these results, some well-known classical inequalities for them are improved, such as Wilker inequality, Huygens inequality, Lazarević inequality and Cusa-Huygens inequality.

Keywords: generalized trigonometric and hyperbolic functions; Wilker inequality; Huygens inequality; Lazarević inequality; Cusa-Huygens inequality
MSC 2010: 26D05; 26D07

1 Introduction

The well-known Wilker inequality for trigonometric functions

(1) sin x x 2 + tan x x > 2 , for x ∈ 0 , π 2

was proposed by Wilker [1] and proved by Sumner et al. [2]. The hyperbolic counterpart of (1) was established in [3] as follows:

(2) sinh x x 2 + tanh x x > 2 , for x ∈ ( 0 , ∞ ) .

A related inequality that is of interest to us is the Huygens inequality [4,5]:

(3) 2 sin x + tan x > 3 x , for x ∈ 0 , π 2 ,

(4) 2 sinh x + tanh x > 3 x , for x ∈ ( 0 , ∞ ) .

The Wilker inequalities (1), (2) and the Huygens inequalities (3), (4) have attracted much interest of many mathematicians. Many generalizations, improvements and refinements of the Wilker inequality and the Huygens inequality can be found in the literature [6,7] and references therein.

In [6, Theorems 5 and 8], inequalities (1)–(4) were improved as

(5) sin p x x 2 + tan p x x > 2 , for x ∈ 0 , π p 2 and p ≥ 2 ;

(6) sinh p x x 2 + tanh p x x > 2 , for x ∈ ( 0 , ∞ ) and 1 < p ≤ 2 ;

(7) 2 sin p x + tan p x > 3 x , for x ∈ 0 , π p 2 and p ≥ 2 ;

(8) 2 sinh p x + tanh p x > 3 x , for x ∈ ( 0 , ∞ ) and 1 < p ≤ 2 ,

where π p = 2 arcsin p 1 = 2 ∫ 0 1 1 ( 1 − t p ) 1 / p d t . For p = 2 , these inequalities coincide with (1)–(4).

In recent years, the following two-sided trigonometric inequality for hyperbolic functions

(9) ( cosh x ) 1 / 3 < sinh x x < cosh x + 2 3 , for x ∈ 0 , π 2

has attracted attention of several research studies. The left inequality of (9) is called the Lazarević inequality, which is obtained in [8]. The right inequality of (9) is the famous Cusa-Huygens inequality, which is obtained in [9], The counterpart of (9) for trigonometric functions

(10) ( cos x ) 1 / 3 < sin x x < cos x + 2 3 , for x ∈ 0 , π 2

is also well known. The left inequality (10) has been proven by Mitrinović [10], while the second one by Cusa and Huygens [4,11]. The aforementioned inequalities have also been obtained in [5].

The generalized trigonometric and hyperbolic functions depending on a parameter p > 1 were studied by Lindqvist in a highly cited paper [12]. Drábek and Manásevich [13] considered a certain ( p , q ) -eigenvalue problem with the Dirichlét boundary condition and found the complete solution to the problem. The solution of a special case is the function sin p,q , which is the first example of the so-called ( p , q ) -trigonometric function. Motivated by the ( p , q ) -eigenvalue problem, Takeuchi [14] has investigated the ( p , q ) -trigonometric functions depending on two parameters in which the case of p = q coincides with the p-function of Lindqvist, and for p = q = 2 they coincide with familiar elementary functions.

In [15], the relations of generalized trigonometric and hyperbolic functions of two parameters with their inverse functions were studied. In [16], some inequalities for ( p , q ) -trigonometric were obtained and a few conjectures for them were posed. Recently, a conjecture posed in [16] was verified in [17]. In [18], the power mean inequality for generalized trigonometric and hyperbolic functions with two parameters was presented.

Motivated by these results on the trigonometric functions, we make a contribution to the subject by showing some Wilker inequalities, Huygens inequalities, Lazarević inequalities and Cusa-Huygens inequalities for the ( p , q ) -trigonometric and hyperbolic functions.

2 Definitions and formulas

For the formulation of our main results, we give the following definitions of ( p , q ) -trigonometric and hyperbolic functions, such as the generalized ( p , q ) -cosine function, the generalized ( p , q ) -tangent function and their inverses, and also the corresponding hyperbolic functions.

For 1 < p , q < ∞ , the increasing function arcsin p , q x : [ 0 , 1 ] → [ 0 , π p , q / 2 ] is defined by

(11) arcsin p , q x = ∫ 0 x 1 ( 1 − t q ) 1 / p d t

and

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

The inverse of arcsin p , q on [ 0 , π p , q / 2 ] is called the generalized ( p , q ) -sine function, denoted by sin p , q : [ 0 , π p , q / 2 ] → [ 0 , 1 ] .

The generalized ( p , q ) -cosine function cos p , q x : [ 0 , π p , q / 2 ] → [ 0 , 1 ] is defined as

(13) cos p , q x = d d x sin p , q x .

If x ∈ [ 0 , π p , q / 2 ] , then by (2.6) in [19]

(14) sin p , q q x + cos p , q p x = 1 .

The generalized ( p , q ) -tangent function tan p , q : ( 0 , π p , q / 2 ) → ( 0 , ∞ ) is defined as

(15) tan p , q x = sin p , q x cos p , q x .

Similarly, for x ∈ ( 0 , ∞ ) , the inverse of the generalized ( p , q ) -hyperbolic sine function [15] is defined by

arcsin h p , q x = ∫ 0 x 1 ( 1 + t q ) 1 / p d t

and also other corresponding ( p , q ) -hyperbolic functions, such as ( p , q ) -hyperbolic cosine and tangent functions, are defined by

cosh p , q x = d d x sinh p , q x , tanh p , q x = sinh p , q x cosh p , q x

for x ∈ [ 0 , ∞ ) , respectively.

The definitions show that

(16) | cosh p , q x | p − | sinh p , q x | q = 1 , x ∈ ( 0 , ∞ ) .

It is clear that all these generalized functions coincide with the classical ones when p = q = 2 .

3 Preliminaries and proofs

In this section, we give three Lemmas needed in the proofs of our main results. First, let us recall the following well-known formulas [15,19]: for p , q ∈ ( 1 , ∞ ) ,

(17) x < sinh p , q x , for x ∈ ( 0 , ∞ ) ,

(18) x < tan p , q x , for x ∈ ( 0 , π p , q / 2 ) ,

(19) x > tanh p , q x , for x ∈ ( 0 , ∞ ) ,

(20) 2 π p , q ≤ sin p , q x x ≤ 1 , for x ∈ 0 , π p , q / 2

and the Jacobsthal inequality [20]

(21) n a n − 1 b ≤ ( n − 1 ) a n + b n , ( a , b > 0 ) .

The following Lemmas will be frequently applied later.

Lemma 1

[15, Lemma 1] [19, Proposition 3.1] For all p , q ∈ ( 1 , ∞ ) , x ∈ ( 0 , π p , q / 2 ) , we have

(22) d d x cos p , q x = − q p ( cos p , q x ) 2 − p ( sin p , q x ) q − 1 ,

(23) d d x tan p , q x = 1 + q p ( sin p , q x ) q ( cos p , q x ) − p ;

and for all x ∈ ( 0 , ∞ ) , we have

(24) d d x cosh p , q x = q p ( cosh p , q x ) 2 − p ( sinh p , q x ) q − 1 ,

(25) d d x tanh p , q x = 1 − q p ( sinh p , q x ) q ( cosh p , q x ) − p .

Lemma 2

  1. For p , q ∈ ( 1 , ∞ ) , the function

    f 1 ( x ) = log ( sin p , q x / x ) log cos p , q x

    is strictly decreasing from ( 0 , π p , q / 2 ) to ( 0 , 1 / ( 1 + q ) ) . In particular, for all p , q ∈ ( 1 , ∞ ) and x ∈ ( 0 , π p , q / 2 ) ,

    (26) cos p , q α x < sin p , q x x < 1

    with the best constant α = 1 / ( 1 + q ) .

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

f 2 ( x ) = log ( sinh p , q x / x ) log cosh p , q x

is strictly increasing from ( 0 , ∞ ) to ( 1 / ( 1 + q ) , 1 ) . In particular, for all p , q ∈ ( 1 , ∞ ) and x ∈ ( 0 , ∞ ) ,

(27) cosh p , q α x < sinh p , q x x < cosh p , q β x

with the best constant α = 1 / ( 1 + q ) and β = 1 .

Proof

  1. Let f 11 ( x ) = log ( sin p , q x / x ) and f 12 ( x ) = log cos p , q x . Clearly, f 11 ( 0 + ) = f 12 ( 0 ) = 0 , by (13) and (22), we have

    f 11 ′ ( x ) f 12 ′ ( x ) = p q tan p , q x − x x cos p , q − p x cos p , q q x = p q f 13 ( x ) f 14 ( x )

    with f 13 ( x ) = tan p , q x − x , f 14 ( x ) = x cos p , q − p x sin p , q q x and f 13 ( 0 ) = f 14 ( 0 ) = 0 . By (14), (22) and (23), we have

    f 13 ′ ( x ) f 14 ′ ( x ) = q p 1 1 + q g 1 ( x )

    with

    g 1 ( x ) = x sin p , q x cos p , q p − 1 x ,

    which is strictly increasing. Using the monotone form of l’Hôpital rule [21, Theorem 1.25], we see that f 1 ( x ) is strictly decreasing. We can easily obtain the limiting values, too. The assertion on f 1 is clear.

  2. Write f 21 ( x ) = log ( sinh p , q x / x ) and f 22 ( x ) = log cosh p , q x , then f 21 ( 0 + ) = f 22 ( 0 ) = 0 . By (24), we have

f 21 ′ ( x ) f 22 ′ ( x ) = p q x − tanh p , q x x cosh p , q − p x sinh p , q q x = p q f 23 ( x ) f 24 ( x )

with f 23 ( x ) = x − tanh p , q x , f 24 ( x ) = x cosh p , q − p x sinh p , q q x and f 23 ( 0 ) = f 24 ( 0 ) = 0 . By differentiation and by (16),

f 23 ′ ( x ) f 24 ′ ( x ) = q p 1 1 + q g 2 ( x )

with

g 2 ( x ) = x sinh p , q x cosh p , q p − 1 x ,

which is strictly decreasing by [15, Lemma 5]. Hence, the function f 2 is strictly increasing by [21, Theorem 1.25]. So the other results follow.□

Remark

The left inequalities of (26) and (27) are Lazarević inequalities.

Lemma 3

For p ≥ q > 2 , the function

g ( x ) = sin p , q q − 2 x cos p , q p − 2 x − sinh p , q q − 2 x cosh p , q p − 2 x

is strictly increasing in ( 0 , π p , q / 2 ) .

Proof

By differentiation, we have

(28) g ′ ( x ) = sin p , q q − 3 x p cos p , q p − 3 x p ( q − 2 ) + q ( p − 2 ) sin p , q q x cos p , q p x − sinh p , q q − 3 x p cosh p , q p − 3 x p ( q − 2 ) − q ( p − 2 ) sinh p , q q x cosh p , q p x .

Case (i). For p ≥ q ≥ 3 .

In this case, by (28), we obtain

g ′ ( x ) ≥ ( q − 2 ) sin p , q q − 3 x cos p , q p − 3 x − ( q − 2 ) sinh p , q q − 3 x cosh p , q p − 3 x = ( q − 2 ) sin p , q q − 3 x cos p , q p − 3 x − sinh p , q q − 3 x cosh p , q p − 3 x ≥ ( q − 2 ) sin p , q q − 3 x cos p , q q − 3 x − sinh p , q q − 3 x cosh p , q q − 3 x = ( q − 2 ) [ tan p , q q − 3 x − tanh p , q q − 3 x ] > 0 ,

which is true by (18) and (19).

Case (ii). For 2 < q ≤ p < 3 .

In this case, by (14), (16), (17) and (28), we have

g ′ ( x ) = sin p , q q − 3 x p q ( p − 2 ) + 2 ( q − p ) cos p , q p x cos p , q 2 p − 3 x − sinh p , q q − 3 x p q ( p − 2 ) + 2 ( q − p ) cosh p , q p x cosh p , q 2 p − 3 x ≥ sinh p , q q − 3 x p q ( p − 2 ) + 2 ( q − p ) cos p , q p x cos p , q 2 p − 3 x − q ( p − 2 ) + 2 ( q − p ) cosh p , q p x cosh p , q 2 p − 3 x = sinh p , q q − 3 x p q ( p − 2 ) 1 cos p , q 2 p − 3 x − 1 cosh p , q 2 p − 3 x + 2 ( q − p ) cos p , q 3 − p x − cosh p , q 3 − p x > 0 ,

which is true since cos p , q x < 1 < cosh p , q x . This completes the proof.□

Lemma 4

For p ≥ q > 1 , the function h ( x ) = cos p , q x ⋅ cosh p , q x is strictly decreasing from ( 0 , π p , q / 2 ) to ( 0 , 1 ) . In particular, for all p ≥ q > 1 and x ∈ ( 0 , π p , q / 2 ) ,

cos p , q x < 1 cosh p , q x .

Proof

By differentiation, we have

h ′ ( x ) = q p cos p , q x ⋅ cosh p , q x sinh p , q q − 1 x cosh p , q p − 1 x − sin p , q q − 1 x cos p , q p − 1 x ≤ q p cos p , q x ⋅ cosh p , q x sinh p , q q − 1 x cosh p , q q − 1 x − sin p , q q − 1 x cos p , q q − 1 x = q p cos p , q x ⋅ cosh p , q x ( tanh p , q q − 1 x − tan p , q q − 1 x ) < 0 .

It is true by (18) and (19), which implies that h is strictly decreasing. Hence, the other conclusion for h is clear.□

4 Main results

Theorem 1

  1. If x ∈ ( 0 , π p , q / 2 ) , p , q > 1 and n ∈ ℕ + , ( n − 1 ) α − q β ≥ 0 , β ≤ 0 , then

    (29) ( n − 1 ) x sin p , q x α + x tan p , q x β > n .

  2. If x ∈ ( 0 , ∞ ) , p , q > 1 and n ∈ ℕ + , ( n − 1 ) α − q β ≤ 0 , β ≤ 0 , then

(30) ( n − 1 ) x sinh p , q x α + x tanh p , q x β > n .

Proof

  1. Taking a = ( x sin p , q x ) α n , b = ( x tan p , q x ) β n in Jacobsthal inequality (21), by (20) and (26) in Lemma 2,

    ( n − 1 ) x sin p , q x α + x tan p , q x β ≥ n x sin p , q x ( n − 1 ) α n x tan p , q x β n = n x sin p , q x ( n − 1 ) α + β n x sin p , q x − β n x tan p , q x β n = n x sin p , q x ( n − 1 ) α + β n ( cos p , q x ) β n > n x sin p , q x ( n − 1 ) α + β n sin p , q x x ( q + 1 ) β n = n x sin p , q x ( n − 1 ) α − q β n > n .

  2. Taking a = ( x sinh p , q x ) α n , b = ( x tanh p , q x ) β n in Jacobsthal inequality (21), the proof of the inequality (30) is similar to that of the proof of inequality (29).□

Remark

  1. Put p = q ≥ 2 , n = 2 , α = − 2 , β = − 1 or put p = q ≥ 2 , n = 3 , α = − 1 , β = − 1 in (29), inequality (29) becomes inequality (5) or (7).

  2. Put 1 < p = q ≤ 2 , n = 2 , α = − 2 , β = − 1 or put 1 < p = q ≤ 2 , n = 3 , α = − 1 , β = − 1 in (30), inequality (30) becomes inequality (6) or (8).

    In particular,

  3. Taking p = q = 2 , n = 2 , α = − 2 , β = − 1 or put p = q = 2 , n = 3 , α = − 1 , β = − 1 in (29) and (30), the inequalities turn into the Wilker inequality (1), (2) or the Huygens inequality (3), (4).

Theorem 2

  1. For x ∈ ( 0 , π p , q / 2 ) , p , q ∈ ( 1 , 2 ] , we have

    (31) sin p , q x x < cos p , q x + q 1 + q ≤ cos p , q x + 2 3 ;

  2. For x ∈ ( 0 , π p , q / 2 ) , p ≥ q ≥ 2 , we have

    (32) sin p , q x x < x sinh p , q x ;

  3. For x ∈ ( 0 , π p , q / 2 ) , p ≥ q > 1 , we have

(33) sin p , q x x > x tan p , q x .

Proof

  1. Let F 1 ( x ) = x ( cos p , q + q ) − ( 1 + q ) sin p , q x . Then by differentiation, we have

    F 1 ′ ( x ) = q − q p x sin p , q q − 1 x cos p , q 2 − p x − q cos p , q x

    and

    F 1 ″ ( x ) = q p cos p , q 3 − p x sin p , q 2 − q x ( q − 1 ) ( tan p , q x − x ) + q ( 2 − p ) p x sin p , q q x cos p , q p x > 0 ,

    which is true by (18) for p , q ∈ ( 1 , 2 ] . Hence, F 1 ( x ) is increasing with F 1 ( 0 ) = 0 , or equivalently,

    x ( cos p , q x + q ) > ( 1 + q ) sin p , q x .

    The right inequality in (31) is clear. It is easy to verify that the left inequality in (31) is true.

  2. Let F 2 ( x ) = x 2 − sin p , q x sinh p , q x , we have

    F 2 ′ ( x ) = 2 x − cos p , q x sinh p , q x − sin p , q x cosh p , q x

    and

    F 2 ″ ( x ) = 2 + q p sin p , q x sinh p , q x sin p , q q − 2 x cos p , q p − 2 x − sinh p , q q − 2 x cosh p , q p − 2 x − 2 cos p , q x cosh p , q x ,

    then F 2 ″ is increasing in x with F 2 ″ ( 0 ) = 0 by Lemmas 3 and 4. Hence, F 2 ′ is increasing in x with F 2 ′ ( 0 ) = 0 , which implies the monotonicity of F 2 with F 2 ( 0 ) = 0 . Therefore, we obtain inequality (32).

  3. Let F 3 ( x ) = sin p , q x tan p , q x − x 2 , then by (14), differentiation gives

F 3 ′ ( x ) = 2 sin p , q x − 2 x + q p sin p , q q + 1 x cos p , q p x ,

F 3 ″ ( x ) = q p sin p , q q x cos p , q 2 p − 1 x ( cos p , q p x + q ) + 2 cos p , q x − 2

and

F 3 ″ ′ ( x ) = q p sin p , q q − 1 x cos p , q 2 p − 2 x p q 2 + p q − q p − 2 p − q p cos p , q p x + q 2 ( 2 p − 1 ) p sin p , q q x cos p , q p x = q p sin p , q q − 1 x cos p , q 2 ( p − 1 ) x ( q 2 + q − 2 ) + 2 − q p sin p , q q + q 2 ( 2 p − 1 ) p sin p , q q x cos p , q p x > 0 .

Hence F 3 ″ ( x ) > F 3 ″ ( 0 ) = 0 , then F 3 ′ is strictly increasing with F 3 ′ ( 0 ) = 0 , and F 3 ( x ) > F 3 ( 0 ) = 0 , inequality (33) follows.□

Theorem 3

  1. For x > 0 , p , q ∈ ( 1 , 2 ] , we have

    (34) sinh p , q x x < cosh p , q x + q 1 + q ;

  2. For x > 0 , q ≥ p ≥ 2 , we have

    (35) sinh p , q x x < cosh p , q x + 2 3 ;

  3. For x > 0 , q ≥ p > 1 , we have

(36) sinh p , q x x > ( p + 1 ) cosh p , q x q cosh p , q x + 1 ,

or equivalently,

(37) tanh p , q x x > p + 1 q cosh p , q x + 1 .

Proof

  1. Let G 1 ( x ) = x ( cosh p , q x + q ) − ( 1 + q ) sinh p , q x , then

    G 1 ′ ( x ) = q + q p x sinh p , q q − 1 x cosh p , q 2 − p x − q cosh p , q x

    and

    G 1 ″ ( x ) = q p cosh p , q 3 − p x sinh p , q 2 − q x ( q − 1 ) ( x − tanh p , q x ) + q ( 2 − p ) p x sinh p , q q x cosh p , q p x > 0 ,

    which is true by (19) for p , q ∈ ( 1 , 2 ] . Hence, G 1 ′ is increasing in x with G 1 ′ ( 0 ) = 0 . So G 1 ( x ) > G 1 ( 0 ) = 0 . Therefore, inequality (34) is obtained.

  2. Let G 2 ( x ) = x ( cosh p , q x + 2 ) − 3 sinh p , q x , then

    G 2 ′ ( x ) = 2 + q p x sinh p , q q − 1 x cosh p , q 2 − p x − 2 cosh p , q x ,

    and by (16), (24), differentiation gives

    G 2 ″ ( x ) = q p sinh p , q q − 2 x cosh p , q p − 3 x ( q − 1 ) x − tanh p , q x + q ( 2 − p ) p x sinh p , q q x cosh p , q p x = q p sinh p , q q − 2 x cosh p , q p − 3 x ( q − 1 ) x 1 − sinh p , q q x cosh p , q p x − tanh p , q x + 2 q − p p x sinh p , q q x cosh p , q p x = q p sinh p , q q − 2 x cosh p , q p − 3 x ( q − 1 ) x cosh p , q p x − tanh p , q x + 2 q − p p x sinh p , q q x cosh p , q p x ≥ q p sinh p , q q − 2 x cosh p , q p − 3 x 2 q − p p x cosh p , q p x − tanh p , q x + 2 q − p p x sinh p , q q x cosh p , q p x = q p sinh p , q q − 2 x cosh p , q p − 3 x 2 q − p p x cosh p , q p x ( 1 + sinh p , q q x ) − tanh p , q x = q p sinh p , q q − 2 x cosh p , q p − 3 x 2 q − p p x − tanh p , q x ≥ q p sinh p , q q − 2 x cosh p , q p − 3 x [ x − tanh p , q x ] ≥ 0 ,

    which is true by (19). So G 2 ′ ( x ) > G 2 ′ ( 0 ) = 0 . Hence, G 2 is increasing in x with G 2 ( x ) > G 2 ( 0 ) = 0 . Then inequality (35) is obtained.

  3. Set G 3 ( x ) = ( p + 1 ) x − tanh p , q x ( q cosh p , q x + 1 ) . Then

G 3 ′ ( x ) = p − q cosh p , q x + q p sinh p , q q x cosh p , q p x ,

and by (16) and (24), differentiation gives

G 3 ″ ( x ) = q 2 p sinh p , q q − 1 x cosh p , q p − 1 x 1 cosh p , q p x − cosh p , q x < 0 .

Hence, G 3 ′ is strictly decreasing with G 3 ′ ( 0 ) = p − q ≤ 0 , and G 3 ( x ) < G 3 ( 0 ) = 0 . Inequalities (36) and (37) are proved.□

Remark

The left inequalities of (31), (34) and (35) are Cusa-Huygens inequalities.

Acknowledgments

This work was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LQ17A010010), the Foundation of the Department of Education of Zhejiang Province (Grant No. Y201840023) and the Natural Science Foundation of China (Grant No. 11171307).

  1. Conflict of interest: The authors have no competing interests.

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Received: 2020-02-01
Revised: 2020-09-30
Accepted: 2020-10-04
Published Online: 2020-12-29

© 2020 Xiaoyan Ma et al., published by De Gruyter

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

Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
https://doi.org/10.1515/math-2020-0096
Keywords for this article
generalized trigonometric and hyperbolic functions; Wilker inequality; Huygens inequality; Lazarević inequality; Cusa-Huygens inequality
Creative Commons
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Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
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