Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Around Jensen's inequality for strongly convex functions   In this paper we use basic properties of strongly convex functions to obtain
new inequalities including Jensen's type and Jensen-Mercer type inequalities.
Applications for special means are pointed out as well. We also give a Jensen's
operator inequality for strongly convex functions. As a corollary, we improve
H\"older-McCarthy inequality under suitable conditions. More precisely we show
that if $Sp\left( A \right)\subset I\subseteq \left( 1,\infty \right)$, then
\[{{\left\langle Ax,x \right\rangle }^{r}}\le \left\langle {{A}^{r}}x,x
\right\rangle -\frac{{{r}^{2}}-r}{2}\left( \left\langle {{A}^{2}}x,x
\right\rangle -{{\left\langle Ax,x \right\rangle }^{2}} \right),\quad r\ge 2\]
and if $Sp\left( A \right)\subset I\subseteq \left( 0,1 \right)$, then
\[\left\langle {{A}^{r}}x,x \right\rangle \le {{\left\langle Ax,x \right\rangle
}^{r}}+\frac{r-{{r}^{2}}}{2}\left( {{\left\langle Ax,x \right\rangle
}^{2}}-\left\langle {{A}^{2}}x,x \right\rangle \right),\quad 0<r<1\] for each
positive operator $A$ and $x\in \mathcal{H}$ with $\left\| x \right\|=1$.

37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int