Majorization inequalities via convex functions

Convex functions have been well studied in the literature for scalars and matrices. However, other types of convex functions have not received the same attention given to the usual convex functions. The main goal of this article is to present matrix inequalities for many types of convex functions, including log-convex, harmonically convex, geometrically convex, and others. The results extend many known results in the literature in this direction. For example, it is shown that if $A,B$ are positive definite matrices and $f$ is a continuous $\sigma\tau$-convex function on an interval containing the spectra of $A,B$, then\begin{align*}\lambda^\downarrow (f(A\sigma B))\prec_w\lambda^\downarrow \left(f(A)\tau f(B)\right),\end{align*}for the matrix means $\sigma,\tau\in\{\nabla_{\alpha},!_{\alpha}\}$ and $\alpha\in[0,1]$. Further, if $\sigma=\sharp_{\alpha}$, then\begin{align*} \lambda^\downarrow \left(f\left(e^{A\nabla_{\alpha}B}\right)\right)\prec_w\lambda^\downarrow \left(f(e^A)\tau f(e^B))\right).\end{align*}Similar inequalities will be presented for two-variable functions too.