Inequalities for Basic Special Functions Using Hölder Inequality

Let $p,q\ge 1$ be two real numbers such that $\frac{1}{p}+\frac{1}{q}=1$, and let $a,b\in \mathbb{R}$ be two parameters defined on the domain of a function, for example, f. Based on the well known Hölder inequality, we propose a generic inequality of the form $\left|f(\frac{a}{p}+\frac{b}{q})\right|\le \left|f\left(a\right){|}^{\frac{1}{p}}\right|f\left(b\right){|}^{\frac{1}{q}},$ and show that many basic special functions, such as the gamma and polygamma functions, Riemann zeta function, beta function and Gauss and confluent hypergeometric functions, satisfy this type of inequality. In this sense, we also present some particular inequalities for the Gauss and confluent hypergeometric functions to confirm the main obtained inequalities.