Home Inequalities for the generalized trigonometric functions with respect to weighted power mean
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Inequalities for the generalized trigonometric functions with respect to weighted power mean

  • Genhong Zhong and Xiaoyan Ma EMAIL logo
Published/Copyright: June 18, 2025
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional p -Laplacian. In this study, the authors investigate some weighted power mean inequalities for the p -generalized trigonometric functions with certain conditions on the parameters p , λ .

Keywords: generalized trigonometric functions; weighted power mean; inequality
MSC 2010: 26E60; 26D05

1 Introduction

In 1995, Lindqvist studied the generalized trigonometric and hyperbolic functions ( p -functions) for a parameter p > 1 [1], and they coincide with elementary functions for p = 2 . These p -functions have been investigated from different points of view (cf. [2,3]). It is worth mentioning that the multiple-angle formulas and double-angle formulas of the generalized trigonometric and hyperbolic functions have been obtained [4–6]. Some inequalities of the Kober and Lazarević type, the Wilker and Cusa type, the Redheffer-type, and the Huygens-type have been established via these functions (cf. [7–11]). Recently, many generalizations, improvements, and refinements of the trigonometric functions have attracted much interest (cf. [12–15]).

For the formulation of our main results, we recall the following definitions of p -trigonometric functions (see [16]), such as the generalized p -sine function, the generalized p -cosine function, the generalized p -tangent function, and their inverses.

For p > 1 , the generalized p-sine function sin p x : [ 0 , π p ⁄ 2 ] → [ 0 , 1 ] is defined by the inverse of the function

arcsin p x = ∫ 0 x 1 ( 1 − t p ) 1 ⁄ p d t .

The generalized p -sine function can be extended to a periodic function ( − ∞ , ∞ ) with the period 2 π p , where π p (a kind of generalized π ) is given by

π p = 2 arcsin p 1 = 2 ∫ 0 1 1 ( 1 − t p ) 1 ⁄ p d t = 2 p B 1 p , 1 − 1 p = 2 π ⁄ p sin ( π ⁄ p ) ,

and

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

are the beta and gamma functions (cf. [17–19]), respectively. The generalized p -sine function sin p and π p appear in the eigenfunction of the Dirichlet problem for one-dimensional p -Laplacian function (cf. [20,21])

(1) d d x d u d x p − 2 d u d x + λ ∣ u ∣ p − 2 u = 0 , u ( 0 ) = u ( 1 ) = 0 .

The eigenvalues of (1) are given as λ n = ( p − 1 ) ( n π p ) p for each n ∈ N , and the corresponding eigenfunction to λ n is u n ( x ) = sin p ( n π p x ) .

The generalized p-cosine function cos p x : [ 0 , π p ⁄ 2 ] → [ 0 , 1 ] is defined by

(2) cos p x = d d x sin p x .

If x ∈ [ 0 , π p ⁄ 2 ] , then

(3) sin p p x + cos p p x = 1 .

The generalized p-tangent function tan p x : ( 0 , π p ⁄ 2 ) → ( 0 , ∞ ) is defined by

(4) tan p x = sin p x cos p x .

Now, we recall the definition of the weighted power mean. For x , y ∈ ( 0 , ∞ ) , ω ∈ ( 0 , 1 ) , and λ ∈ R , the weighted power mean M λ of order λ is defined by

(5) M λ = M λ ( ω , x , y ) = [ ω x λ + ( 1 − ω ) y λ ] 1 ⁄ λ , λ ≠ 0 , x ω y 1 − ω , λ = 0 .

As special cases, we have

M λ ( 1 ⁄ 2 , x , y ) = H λ ( x , y ) , M 1 = M 1 ( 1 ⁄ 2 , x , y ) = A ( x , y ) , M 0 = M 0 ( 1 ⁄ 2 , x , y ) = G ( x , y ) , M − 1 = M − 1 ( 1 ⁄ 2 , x , y ) = H ( x , y ) ,

where

H λ ( x , y ) = x λ + y λ 2 1 ⁄ λ , λ ≠ 0 , x y , λ = 0 ,

A ( x , y ) = x + y 2 , G ( x , y ) = x y , H ( x , y ) = 2 x y x + y

are the Hölder mean of order λ , the Arithmetic, Geometric, harmonic means of x and y, respectively. The main properties of the weighted power mean were given in [22–25]. And some power mean inequalities for the complete and the generalized elliptic integrals (cf. [26–29]) and the other special functions (cf. [30,31]) have been found.

Let I ⊆ ( 0 , ∞ ) be an interval and f : I → ( 0 , ∞ ) be a continuous function. Then, f is said to be M a , b -convex (concave) on I if the inequality

(6) f ( M a ( x , y ) ) ≤ ( ≥ ) M b ( f ( x ) , f ( y ) )

holds for all x , y ∈ I . Actually, M 0,0 -convex (concave) is equivalent to G G -convex (concave), M 1,1 -convex (concave) is equivalent to A A -convex (concave), and so on.

In the past few years, numerous authors have studied the M a , b -convexity (concavity) properties for the generalized trigonometric functions (cf. [32–36]). For example, in 2012, Bhayo and Vuorinen raised a conjecture that the generalized sine function with two parameters sin p , q x , ( sin p , p ( x ) = sin p x ) is GG-convex on ( 0 , 1 ) for p , q ∈ ( 1 , ∞ ) [34]. In 2013, Jiang et al. verified this conjecture [35]. Later, Bhayo proved that the convexity/concavity properties of the generalized p -sine function sin p x , the generalized p -cosine function cos p x , and the generalized p -tangent function tan p x with respect to the Hölder mean for p ∈ ( 1 , ∞ ) and λ ∈ [ 1 , ∞ ) , respectively [36].

In this study, we show that the generalized p -sine function, the generalized p -cosine function, and the generalized p -tangent function are M λ , λ -convex/concave, and we obtain the weighted power mean inequalities for them. Our main results are the following theorems.

Theorem 1.1

For λ ≥ − p , p > 1 , and ω ∈ ( 0 , 1 ) , the inequality

(7) sin p ( M λ ( ω , x , y ) ) ≥ M λ ( ω , sin p x , sin p y )

holds for all x , y ∈ ( 0 , π p ⁄ 2 ) , with equality holds if and only if x = y .

Theorem 1.2

For λ ≤ p ⁄ 2 , p ≥ 2 , and ω ∈ ( 0 , 1 ) , the inequality

(8) cos p ( M λ ( ω , x , y ) ) ≥ M λ ( ω , cos p x , cos p y )

holds for all x , y ∈ ( 0 , π p ⁄ 2 ) , and for λ ≥ p , inequality (8) is reversed, with equality holds if and only if x = y .

Theorem 1.3

For λ ≥ 1 − p , p > 1 , and ω ∈ ( 0 , 1 ) , inequality

(9) tan p ( M λ ( ω , x , y ) ) ≤ M λ ( ω , tan p x , tan p y )

holds for all x , y ∈ ( 0 , π p ⁄ 2 ) , and for λ ≤ − p , inequality (9) is reversed, with equality holds if and only if x = y .

2 Preliminaries

In this section, we give five lemmas needed in the proofs of our main results. First, let us recall the following well-known formulas (cf. [3]): For all p ∈ ( 1 , ∞ ) , x ∈ ( 0 , π p ⁄ 2 ) ,

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

(11) d d x tan p x = 1 + tan p p x .

Lemma 2.1

For p > 1 , we have

  1. The function f 1 ( x ) = sin p x ⁄ x is strictly decreasing from ( 0 , π p ⁄ 2 ) to ( 2 ⁄ π p , 1 ) .

  2. The function f 2 ( x ) = tan p x ⁄ x is strictly increasing from ( 0 , π p ⁄ 2 ) to ( 1 , ∞ ) .

  3. The function f 3 ( x ) = ( tan p x − x ) ⁄ ( x tan p p x ) is decreasing from ( 0 , π p ⁄ 2 ) to ( 0 , 1 ⁄ ( 1 + p ) ) .

  4. The function f 4 ( x ) = tan p x ⋅ cos p p x + p x is concave and increasing from ( 0 , π p ⁄ 2 ) to ( 0 , p π p ⁄ 2 ) .

Proof

Part (1), part (2), and part (3) follow from [3, Lemma 3.32 (1), (2)] and the proof of [3, Theorem 3.6], respectively.

(4) By (3), (10), and (11), differentiation gives

f 4 ′ ( x ) = p + 1 − p sin p p x = p cos p p x + 1 ,

which is decreasing from ( 0 , π p ⁄ 2 ) to ( 1 , p + 1 ) . Hence, the monotonicity and convexity properties of f 4 follow. Clearly, f 4 ( 0 ) = 0 and f 4 ( π p ⁄ 2 ) = p π p ⁄ 2 .□

Lemma 2.2

For p > 1 and λ ∈ R , the function f λ is defined on ( 0 , π p ⁄ 2 ) by

f λ ( x ) = sin p x x λ − 1 ⋅ cos p x .

Then, the following statements are true:

  1. If λ ≥ − p , the function f λ ( x ) is decreasing from ( 0 , π p ⁄ 2 ) to ( 0 , 1 ) .

  2. If λ < − p , then there exists a unique x 0 ∈ ( 0 , π p ⁄ 2 ) such that f λ ( x ) is increasing on ( 0 , x 0 ] and decreasing on [ x 0 , π p ⁄ 2 ) .

Proof

Clearly, f λ ( 0 + ) = 1 and f λ ( π p ⁄ 2 ) = 0 . We now prove the result for f λ by investigating three cases.

Case i. λ ≥ 1 .

The monotonicity of f λ ( x ) follows from Lemma 2.1 (1).

Case ii. − p ≤ λ < 1 .

By logarithmic differentiation in x , we obtain

(12) f λ ′ ( x ) f λ ( x ) = ( λ − 1 ) cos p x sin p x − 1 x − cos p 1 − p x ⋅ sin p p − 1 x = ( 1 − λ ) tan p x − x x tan p x − tan p p − 1 x .

Therefore,

(13) f λ ′ ( x ) f λ ( x ) tan p p − 1 x = ( 1 − λ ) tan p x − x x tan p p x − 1 = ( 1 − λ ) f 3 ( x ) − 1 ,

where f 3 ( x ) is in Lemma 2.1 (3). Let φ 1 ( x ) = ( 1 − λ ) f 3 ( x ) − 1 , then φ 1 is decreasing from ( 0 , π p ⁄ 2 ) to ( − 1 , − ( λ + p ) ⁄ ( 1 + p ) ) . For x ∈ ( 0 , π p ⁄ 2 ) and λ ∈ [ − p , 1 ) , we obtain f ′ ( x ) < 0 by (13) and Lemma 2.1 (3). Hence, the monotonicity property of f λ follows.

Case iii. λ < − p .

This case shows that φ 1 has a unique zero x 0 ∈ ( 0 , π p ⁄ 2 ) , which is the solution to the equation φ 1 ( x ) = 0 such that f λ ( x ) is increasing on ( 0 , x 0 ] and decreasing on [ x 0 , π p ⁄ 2 ) . Hence, we obtain the piecewise monotonicity property of f λ .

Consequently, the result for f λ follows from the investigations in Cases i–iii.□

Lemma 2.3

For p ≥ 2 and λ ∈ R , the function g λ is defined on ( 0 , π p ⁄ 2 ) by

g λ ( x ) = cos p λ + 1 − p x ⋅ sin p p − 1 x x λ − 1 .

Then, the following statements are true:

  1. If λ ≤ p ⁄ 2 , the function g λ ( x ) is increasing from ( 0 , π p ⁄ 2 ) to ( 0 , ∞ ) .

  2. If λ ≥ p , the function g λ ( x ) is decreasing from ( 0 , π p ⁄ 2 ) to ( 0 , ∞ ) .

Proof

We now prove the result for g λ by investigating three cases.

Case i. λ ≤ 1 .

The function g λ ( x ) can be written as

g λ ( x ) = x cos p x 1 − λ ⋅ sin p p − 1 x cos p p − 2 x ,

which is a product of two positive and increasing functions. Hence, the monotonicity property of g λ is obtained.

Case ii. 1 < λ ≤ p ⁄ 2 .

By logarithmic differentiation in x , we obtain

(14) x tan p x ⋅ g λ ′ ( x ) g λ ( x ) = ( p − λ − 1 ) x tan p p x + ( p − 1 ) x − ( λ − 1 ) tan p x = φ 2 ( x ) ,

where φ 2 ( x ) = ( p − λ − 1 ) x tan p p x + ( p − 1 ) x − ( λ − 1 ) tan p x , then

φ 2 ′ ( x ) = ( p − 2 λ ) tan p p x + p ( p − λ − 1 ) x tan p p − 1 x ( 1 + tan p p x ) + ( p − λ ) ,

then φ ′ ( x ) > 0 for 1 < λ ≤ p ⁄ 2 . Hence, φ 2 is increasing with φ 2 ( x ) > φ 2 ( 0 ) = 0 . So we obtain that g λ ′ ( x ) > 0 by (14) for x ∈ ( 0 , π p ⁄ 2 ) and λ ∈ ( 1 , p ⁄ 2 ] . Hence, the monotonicity property of g λ follows.

Hence, g λ is increasing for λ ≤ p ⁄ 2 from Cases i and ii. Clearly, g λ ( 0 + ) = 0 and g λ ( π p ⁄ 2 ) = ∞ .

Case iii. λ ≥ p .

By (14), we obtain

(15) 1 tan p p − 1 x g λ ′ ( x ) g λ ( x ) = ( p − λ − 1 ) + p − 1 tan p p x + 1 − λ x tan p p − 1 x = ( p − λ − 1 ) + p − λ tan p p x + ( 1 − λ ) tan p x − x x tan p p x = ( p − λ − 1 ) + p − λ tan p p x + ( 1 − λ ) f 3 ( x ) ,

where f 3 is in Lemma 2.1 (3), and g λ ′ is negative for λ ≥ p by Lemma 2.1 (3) and (15). Hence, the monotonicity property of g λ is obtained.

Clearly, g λ ( 0 + ) = ∞ and g λ ( π p ⁄ 2 ) = 0 .

Consequently, the result for g λ follows from the investigations in Cases i–iii.□

Lemma 2.4

For p > 1 and λ ∈ R , the function h λ is defined on ( 0 , π p ⁄ 2 ) by

h λ ( x ) = tan p λ − 1 x x λ − 1 cos p p x .

Then, the following statements are true:

  1. If λ ≥ 1 − p , then the function h λ ( x ) is increasing from ( 0 , π p ⁄ 2 ) to ( 1 , ∞ ) .

  2. If λ ≤ − p , then the function h λ ( x ) is decreasing from ( 0 , π p ⁄ 2 ) to ( 0 , 1 ) .

  3. If − p < λ < 1 − p , then there exists a unique x 1 ∈ ( 0 , π p ⁄ 2 ) such that h λ ( x ) is increasing on ( 0 , x 1 ] and decreasing on [ x 1 , π p ⁄ 2 ) .

Proof

We now prove the result for h λ by investigating three cases.

Case i. λ ≥ 1 .

The function h λ ( x ) can be written as

h λ ( x ) = tan p x x λ − 1 ⋅ 1 cos p p x ,

which is a product of two positive and increasing functions. Hence, the monotonicity property of h λ is obtained. Clearly, h λ ( 0 ) = 1 , h λ ( π p ⁄ 2 ) = ∞ .

Case ii. 1 − p ≤ λ < 1 or λ ≤ − p .

By logarithmic differentiation in x , we obtain

(16) 1 tan p p − 1 x ⋅ h λ ′ ( x ) h λ ( x ) = ( λ + p − 1 ) + ( 1 − λ ) tan p x − x x tan p p x = ( λ + p − 1 ) + ( 1 − λ ) f 3 ( x ) ,

where f 3 is in Lemma 2.1 (3). Let φ 3 ( x ) = ( λ + p − 1 ) + ( 1 − λ ) f 3 ( x ) , then φ 3 is decreasing from ( 0 , π p ⁄ 2 ) to ( λ + p − 1 , p ( λ + p ) ⁄ ( 1 + p ) ) for λ < 1 by Lemma 2.1 (3). Hence, h λ ′ is nonnegative for 1 − p ≤ λ < 1 and h λ ′ is nonpositive for λ ≤ − p by Lemma 2.1 (3) and (16). Hence, h λ is increasing for 1 − p ≤ λ < 1 and decreasing for λ ≤ − p .

Clearly, h λ ( 0 ) = 1 , h λ ( π p ⁄ 2 ) = ∞ for λ ≥ 1 − p , and h λ ( 0 ) = 1 , h λ ( π p ⁄ 2 ) = 0 for λ ≤ − p .

Case iii. − p < λ < 1 − p .

In this case, φ 3 has a unique zero x 1 ∈ ( 0 , π p ⁄ 2 ) , which is the solution to the equation φ 3 ( x ) = 0 such that h λ ( x ) is increasing on ( 0 , x 1 ] and decreasing on [ x 1 , π p ⁄ 2 ) by (16). Hence, we obtain the piecewise monotonicity property of h λ .

Consequently, the result for h λ follows from the investigations in Cases i–iii.□

Lemma 2.5

For ω ∈ ( 0 , 1 ) , p ∈ ( 1 , ∞ ) , x , y ∈ ( 0 , π p ⁄ 2 ) ,

(17) sin p ( x ω y 1 − ω ) ≥ sin p ω x ⋅ sin p 1 − ω y ,

(18) cos p ( x ω y 1 − ω ) ≥ cos p ω x ⋅ cos p 1 − ω y ,

(19) tan p ( x ω y 1 − ω ) ≤ tan p ω x ⋅ tan p 1 − ω y ,

with equalities holding if and only if x = y .

Proof

We may assume that x ≤ y , define

g 1 ( x , y ) = sin p ( x ω y 1 − ω ) sin p ω x ⋅ sin p 1 − ω y , g 2 ( x , y ) = cos p ( x ω y 1 − ω ) cos p ω x ⋅ cos p 1 − ω y ,  and   g 3 ( x , y ) = tan p ( x ω y 1 − ω ) tan p ω x ⋅ tan p 1 − ω y ,

respectively. Let t 1 = x ω y 1 − ω , then ∂ t 1 ⁄ ∂ x = ω ( y ⁄ x ) 1 − ω , ∂ t 1 ⁄ ∂ y = ( 1 − ω ) ( x ⁄ y ) ω . If y > x , then t 1 > x . By logarithmic differentiation in x , we have

(20) 1 g 1 ( x , y ) ∂ g 1 ∂ x = cos p ( t 1 ) sin p ( t 1 ) ⋅ ω y x 1 − ω − ω cos p x sin p x = ω tan p ( t 1 ) ⋅ t 1 x − ω tan p x = ω x t 1 tan p ( t 1 ) − x tan p x ,

and similarly,

(21) 1 g 2 ( x , y ) ∂ g 2 ∂ x = ω x [ x tan p p − 1 x − t 1 tan p p − 1 ( t 1 ) ] ,

(22) 1 g 3 ( x , y ) ∂ g 3 ∂ x = ω x t 1 sin p ( t 1 ) cos p p − 1 ( t 1 ) − x sin p x cos p p − 1 x .

Hence, it follows from Lemma 2.1 (2) that ∂ g 1 ⁄ ∂ x < 0 . By Lemma 2.1 (1), we obtain ∂ g 3 ⁄ ∂ x > 0 . On the other hand, it is obvious that ∂ g 2 ⁄ ∂ x < 0 .

Consequently,

g 1 ( x , y ) ≥ g 1 ( y , y ) = 1 , g 2 ( x , y ) ≥ g 2 ( y , y ) = 1 , and g 3 ( x , y ) ≤ g 3 ( y , y ) = 1 .

Thus, we obtain inequalities (17), (18), and (19) for x , y ∈ ( 0 , π p ⁄ 2 ) with x ≠ y . Equalities are valid if and only if x = y .□

3 Proofs of the main results

In this section, we prove the main results.

Proof of Theorem 1.1

If λ = 0 , inequality (7) has been proved by inequality (17) in Lemma 2.5. So we only need to prove inequality (7) for λ ≥ − p and λ ≠ 0 . We may assume that x ≤ y . Define

F ( x , y ) = sin p λ ( M λ ( ω , x , y ) ) − ω sin p λ x − ( 1 − ω ) sin p λ y .

Let t 2 = M λ ( ω , x , y ) , then ∂ t 2 ⁄ ∂ x = ω ( x ⁄ t 2 ) λ − 1 and ∂ t 2 ⁄ ∂ y = ( 1 − ω ) ( y ⁄ t 2 ) λ − 1 . If y > x , then t 2 > x . By differentiation in x , we have

(23) ∂ F ∂ x = λ sin p λ − 1 ( t 2 ) ⋅ cos p ( t 2 ) ⋅ ω x t 2 λ − 1 − ω λ ⋅ sin p λ − 1 x cos p x = ω λ ⋅ x λ − 1 sin p λ − 1 ( t 2 ) cos p ( t 2 ) t 2 λ − 1 − sin p λ − 1 ( x ) cos p ( x ) x λ − 1 = ω λ ⋅ x λ − 1 [ f λ ( t 2 ) − f λ ( x ) ] ,

where f λ is in Lemma 2.2.

Next we divide the proof into the following two cases:

Case i. − p ≤ λ < 0 .

By Lemma 2.2 and (23), we know that ∂ F ⁄ ∂ x > 0 . Hence, F ( x , y ) < F ( y , y ) = 0 . Thus, we obtain

(24) sin p λ ( M λ ( ω , x , y ) ) < ω sin p λ x + ( 1 − ω ) sin p λ y

for x , y ∈ ( 0 , π p ⁄ 2 ) with x ≠ y . From (24), we can easily obtain that inequality (7) holds for all x , y ∈ ( 0 , π p ⁄ 2 ) and − p ≤ λ < 0 , and equality is valid if and only if x = y .

Case ii. λ > 0 .

By Lemma 2.2 and (23), we know that ∂ F ⁄ ∂ x < 0 . Hence, F ( x , y ) > F ( y , y ) = 0 . Thus, we obtain

(25) sin p λ ( M λ ( ω , x , y ) ) > ω sin p λ x + ( 1 − ω ) sin p λ y

for x , y ∈ ( 0 , π p ⁄ 2 ) with x ≠ y . From (25), we can easily obtain that inequality (7) holds for all x , y ∈ ( 0 , π p ⁄ 2 ) and λ > 0 , and equality (7) is valid if and only if x = y . The proof of Theorem 1.1 is complete.□

Proof of Theorem 1.2

If λ = 0 , inequality (8) has been proved by the inequality (18) in Lemma 2.5. So we only need to prove inequality (8) for λ ≤ p ⁄ 2 and λ ≥ p . We may assume that x ≤ y . Define

G ( x , y ) = cos p λ ( M λ ( ω , x , y ) ) − ω cos p λ x − ( 1 − ω ) cos p λ y .

By differentiation in x , we have

(26) ∂ G ∂ x = − ω λ cos p λ − p + 1 ( t 2 ) ⋅ sin p p − 1 ( t 2 ) ⋅ x t 2 λ − 1 + ω λ ⋅ cos p λ − p + 1 x sin p p − 1 x = ω λ ⋅ x λ − 1 cos p λ + 1 − p x sin p p − 1 x x λ − 1 − cos p λ + 1 − p ( t 2 ) sin p p − 1 ( t 2 ) t 2 λ − 1 = ω λ ⋅ x λ − 1 [ g λ ( x ) − g λ ( t 2 ) ] ,

where g λ is in Lemma 2.3.

Next we divide the proof into the following two cases:

Case i. 0 < λ ≤ p ⁄ 2 .

By Lemma 2.3 and (26), we know that ∂ G ⁄ ∂ x < 0 . Hence, G ( x , y ) > G ( y , y ) = 0 . Thus, we obtain

(27) cos p λ ( M λ ( ω , x , y ) ) > ω cos p λ x + ( 1 − ω ) cos p λ y

for x , y ∈ ( 0 , π p ⁄ 2 ) with x ≠ y . From (27), we can easily obtain that inequality (8) holds for all x , y ∈ ( 0 , π p ⁄ 2 ) and 0 < λ ≤ p ⁄ 2 , and equality is valid if and only if x = y .

Case ii. λ < 0 or λ ≥ p .

By Lemma 2.3 and (26), we know that ∂ G ⁄ ∂ x > 0 . Hence, G ( x , y ) < G ( y , y ) = 0 . Thus, we obtain

(28) cos p λ ( M λ ( ω , x , y ) ) < ω cos p λ x + ( 1 − ω ) cos p λ y

for x , y ∈ ( 0 , π p ⁄ 2 ) with x ≠ y . From (28), we can easily obtain that inequality (8) is reversed for λ ≥ p and it holds for λ < 0 , and equality (8) is valid if and only if x = y . The proof of Theorem 1.2 is complete.□

Proof of Theorem 1.3

If λ = 0 , inequality (9) has been proved by inequality (19) in Lemma 2.5. So we only need to prove the inequality (9) for 1 − p ≤ λ < 0 , λ > 0 and λ ≤ − p . We may assume that x ≤ y . Define

H ( x , y ) = tan p λ ( M λ ( ω , x , y ) ) − ω tan p λ x − ( 1 − ω ) tan p λ y .

By differentiation in x , we have

(29) ∂ H ∂ x = ω λ tan p λ − 1 ( t 2 ) ⋅ ( 1 + tan p p ( t 2 ) ) ⋅ x t 2 λ − 1 − ω λ ⋅ tan p λ − 1 x ( 1 + tan p p x ) = ω λ ⋅ x λ − 1 tan p λ − 1 ( t 2 ) t 2 λ − 1 cos p p ( t 2 ) − tan p λ − 1 x x λ − 1 cos p p x = ω λ ⋅ x λ − 1 [ h λ ( t 2 ) − h λ ( x ) ] ,

where h λ is in Lemma 2.4.

Next we divide the proof into the following two cases:

Case i. 1 − p ≤ λ < 0 .

By Lemma 2.4 (1) and (29), we know that ∂ H ⁄ ∂ x < 0 . Hence, H ( x , y ) > H ( y , y ) = 0 . Thus, we obtain

(30) tan p λ ( M λ ( ω , x , y ) ) > ω tan p λ x + ( 1 − ω ) tan p λ y

for x , y ∈ ( 0 , π p ⁄ 2 ) with x ≠ y . From (30), we can easily obtain that the inequality (9) holds for all x , y ∈ ( 0 , π p ⁄ 2 ) and 1 − p ≤ λ < 0 , and equality (9) is valid if and only if x = y .

Case ii. λ ≤ − p or λ > 0 .

By Lemma 2.4 and (29), we know that ∂ H ⁄ ∂ x > 0 . Hence, H ( x , y ) < H ( y , y ) = 0 . Thus, we obtain

(31) tan p λ ( M λ ( ω , x , y ) ) < ω tan p λ x + ( 1 − ω ) tan p λ y

for x , y ∈ ( 0 , π p ⁄ 2 ) with x ≠ y . From (31), we can easily obtain that inequality (9) is reversed for λ ≤ − p and equality (9) holds for λ > 0 , and equality (9) is valid if and only if x = y . The proof of Theorem 1.3 is complete.□

Acknowledgements

The authors are grateful for the reviewer’s valuable comments that improved the manuscript.

  1. Funding information: This research is supported by Shaoxing Basic Public Welfare Plan Project (2024A11021), the Second Batch of Undergraduate Teaching Reform Filing Project in Zhejiang Province for the “14th Five-Year Plan” (JGBA2024139), the 14th Five-Year Teaching Reform Project for Ordinary Undergraduate Universities in Zhejiang Province (jg20220747).

  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.

  4. Ethical approval: The conducted research is not related to either human or animals use.

  5. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2024-02-20
Revised: 2024-11-24
Accepted: 2024-12-12
Published Online: 2025-06-18

© 2025 the author(s), 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/dema-2025-0103
Keywords for this article
generalized trigonometric functions; weighted power mean; inequality
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Articles in the same Issue

  1. Research Articles
  2. On approximation by Stancu variant of Bernstein-Durrmeyer-type operators in movable compact disks
  3. Circular n,m-rung orthopair fuzzy sets and their applications in multicriteria decision-making
  4. Grand Triebel-Lizorkin-Morrey spaces
  5. Coefficient estimates and Fekete-Szegö problem for some classes of univalent functions generalized to a complex order
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  12. Fixed point results for contractions of polynomial type
  13. Estimation for spatial semi-functional partial linear regression model with missing response at random
  14. Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 < η ≤ 1 < ζ ≤ 2
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  16. Rate of pole detection using Padé approximants to polynomial expansions
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