MONOTONICITY AND LOGARITHMIC CONVEXITY OF THREE FUNCTIONS INVOLVING EXPONENTIAL FUNCTION

Abstract

In this note, an alternative proof and extensions are provided for the following conclusions in [6, Theorem 1 and Theorem 3]: The functions $\frac1{x^2}-\frac{e^{-x}}{(1-e^{-x})^2}\;and\;\frac1{t}-\frac1{e^t-1}$ are decreasing in (0, ${\infty}$) and the function $\frac{t}{e^{at}-e^{(a-1)t}}$ for a $a{\in}\mathbb{R}\;and\;t\;{\in}\;(0,\;{\infty})$ is logarithmically concave.

Keywords

• alternative proof;
• monotonicity;
• logarithmic convexity;
• exponential function;
• extension;
• Hermite-Hadamard's integral inequality;
• inequality;
• power series expansion