# Some new inequalities for Generalized Mathieu type series and Riemann zeta functions

## TL;DR

This paper introduces new inequalities, monotonicity, and log-convexity results for Mathieu type series and the Riemann zeta function, along with novel integral representations.

## Contribution

It provides new inequalities and integral representations for Mathieu type series and the Riemann zeta function, enhancing understanding of their properties.

## Key findings

- Turán type inequalities established
- Monotonicity and log-convexity results proved
- New Laplace integral representations derived

## Abstract

Our aim in this paper is to show some new inequalities for Mathieu's type series and Riemann zeta function. In particular, some Tur\'an type inequalities, some monotonicity and log-convexity results for these special functions are given. New Laplace type integral representations for Mathie type series and Riemann zeta function are also presented.