Optimal bounds for a Toader-type mean in terms of one-parameter quadratic and contraharmonic means

Abstract

In this paper, we present the best possible Toader mean bounds of arithmetic and quadratic means by the one-parameter quadratic and contraharmonic means. As applications in engineering and technology, we find new bounds for the complete elliptic integral of the second kind.

Inequalities Involving Conformable Approach for Exponentially Convex Functions and Their Applications

“…A bivariate function Ω: (0, ∞) × (0, ∞) ⟶ (0, ∞) is said to be a mean if min a, b { } ≤ Ω(a, b) ≤ max a, b { } for all a, b ∈ (0, ∞). Recently, the bivariate means have been the subject of intensive research [63][64][65][66][67][68][69][70][71][72][73][74][75]; in particular, many remarkable inequalities and properties for the bivariate means and their related special functions can be found in the literature [76][77][78][79][80][81][82][83][84][85].…”

Sharp bounds for Neuman means in terms of two-parameter contraharmonic and arithmetic mean

“…Author details 1 School of Continuing Education, Huzhou Vocational & Technical College, Huzhou, China. 2 College of Mathematics and Econometrics, Hunan University, Changsha, China.…”

“…Let x, y > 0. Then the arithmetic mean A(x, y), quadratic mean Q(x, y) [1], contraharmonic mean C(x, y) [2,3], and Schwab-Borchardt mean SB(x, y) [4] are given by…”

Optimal inequalities for a Toader-type mean by quadratic and contraharmonic means

“…In [8], the authors found the best possible parameters λ 1 , µ 1 , λ 2 , µ 2 ∈ (1/2, 1) such that the double inequalities…”

“…Motivated by inequality (1.5) and the results of [8], it is natural to ask what are the best possible parameters α i , β i (i = 1, 2, 3) and α 4 , β 4 ∈ (1/2, 1) such that the double inequalities…”