Complete (p,q)-elliptic integrals with application to a family of means

Abstract

Abstract. The complete elliptic integrals are generalized by using the generalized trigonometric functions with two parameters. As an application of the integrals, an alternative proof of a result for a family of means by Bhatia and Li, which involves the logarithmic mean and the arithmetic-geometric mean, is given. Moreover, it is shown that a particular relation holds for the generalized integrals.Mathematics subject classification (2010): 33E05, 33C75, 34L10.

Fractional integral inequalities for generalized-$$\mathbf{m }$$-$$((h_{1}^{p},h_{2}^{q});(\eta _{1},\eta _{2}))$$-convex mappings via an extended generalized Mittag–Leffler function

“…In [1], properties of the extended generalized Mittag-Leffler function are studied in details, and it is given that E γ,δ,k,c μ,α,l (t; p) is absolutely convergent for k < δ + R(μ). For other recent results see [15,37,38,46]. Let S be the sum of series of absolute terms of E γ,δ,k,c μ,α,l (t; p), then we have E γ,δ,k,c μ,α,l (t; p) ≤ S. We will use this property of extended generalized Mittag-Leffler function in sequel.…”

“…The trapezium type inequality has remained an area of great interest due to its wide applications in the field of mathematical analysis. For other recent results which generalize, improve and extend the inequality (1.1) through various classes of convex functions interested readers are referred to [2,[4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][25][26][27][28][29][30][31][33][34][35][36]41,44,45]. where β p is the generalized beta function defined by (1−t) dt (1.3) and (c) nk is the Pochhammer symbol defined as (c + nk) (c) .…”

On a function involving generalized complete (p, q)-elliptic integrals

“…is known as the generalized sine function with two parameters p, q > 1 in the literature (see, [9,10,14,21,24,[28][29][30]), and defined as the inverse function of…”

Functional Inequalities for Generalized Complete Elliptic Integrals with Two Parameters

“…Based on the above definitions, we simply introduce generalized elliptic integrals defined by Takeuchi [10] with two parameters. For all , ∈ (1, ∞) and ∈ (0, 1), the complete ( , )-elliptic integrals of the first and second kinds [11] are defined by…”