On two new means of two arguments III

In this paper authors establish the two sided inequalities for the following two new means $$X=X(a,b)=Ae^{G/P-1},\quad Y=Y(a,b)=Ge^{L/A-1}.$$ As well as many other well known inequalities involving the identric mean $I$ and the logarithmic mean are refined from the literature as an application.Comment: 14. arXiv admin note: substantial text overlap with arXiv:1509.0197

Bounds for the Ratios of Differences of Power Means in Two Arguments

Using methods from classical analysis, sharp bounds for the ratio of differences of Power Means are obtained. Our results generalize and extend previous ones due to S. Wu(2005), and to S. Wu and L. Debnath.Comment: 1 figur

A new way to prove L'Hospital Monotone Rules with applications

Let $-\infty \leq a<b\leq \infty$. Let $f$ and $g$ be differentiable functions on $(a,b)$ and let $g^{\prime }\neq 0$ on $(a,b)$. By introducing an auxiliary function $H_{f,g}:=\left( f^{\prime }/g^{\prime }\right) g-f$, we easily prove L'Hoipital rules for monotonicity. This offer a natural and concise way so that those rules are easier to be understood. Using our L'Hospital Piecewise Monotone Rules (for short, LPMR), we establish three new sharp inequalities for hyperbolic and trigonometric functions as well as bivariate means, which supplement certain known results.Comment: 19 page

Functional inequalities for modified Bessel functions

In this paper our aim is to show some mean value inequalities for the modified Bessel functions of the first and second kinds. Our proofs are based on some bounds for the logarithmic derivatives of these functions, which are in fact equivalent to the corresponding Tur\'an type inequalities for these functions. As an application of the results concerning the modified Bessel function of the second kind we prove that the cumulative distribution function of the gamma-gamma distribution is log-concave. At the end of this paper several open problems are posed, which may be of interest for further research.Comment: 14 page

Generalized Elliptic Integrals and the Legendre M-function

We study monotonicity and convexity properties of functions arising in the theory of elliptic integrals, and in particular in the case of a Schwarz-Christoffel conformal mapping from a half-plane to a trapezoid. We obtain sharp monotonicity and convexity results for combinations of these functions, as well as functional inequalities and a linearization property.Comment: 28 page