New inequalities for $(p,h)$-convex functions for $\tau$-measurable operators

Abstract

The main goal of this article is to present new inequalities for $(p,h)$-convex and $(p,h)$ log-convex functions for a non-negative super-multiplicative and super-additive function $h$. Our first main result will be
\begin{align*}
h^\lambda\left(\frac{v}{\mu}\right) \leq \frac{(h(1-v) f(a)+h(v) f(b))^\lambda-f^\lambda\left[((1-v)a^p+v b^p)^{\frac{1}{p}}\right]}{(h(1-\mu) f(a)+h(\mu) f(b))^\lambda-f^\lambda\left[((1-\mu)a^p+\mu b^p)^{\frac{1}{p}}\right]}\leq h^\lambda\left(\frac{1-v}{1-\mu}\right) ,
\end{align*}
for the positive $(p,h)$-convex function $f,$ when $\lambda\geq 1,$ $p\in\mathbb{R}\backslash\{0\}$ and $0<v\leq \mu<1.$ Which gives a generalization of an important result due to M. Sababheh [Linear Algebra Appl. \textbf{506} (2016), 588--602]. As applications of our results, we present many inequalities for the trace, and the symmetric norms for $\tau$-measurable operators.