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Some inequalities related to Jensen and Ostrowski inequalities for general Lebesgue integral are obtained. Applications for f-divergence measure are provided as well.

Some inequalities related to Jensen and Ostrowski inequalities for general Lebesgue integral of differentiable functions whose derivatives in absolute value are h-convex are obtained. Applications for f-divergence measure are provided as well.

We establish in this paper some Jensen’s type inequalities for functions defined by power series with nonnegative coefficients. Applications for functions of selfadjoint operators on complex Hilbert spaces are provided as well.

Some new bounds for the Čebyšev functional in terms of the Lebesgue norms $${∥f-\frac{1}{b-a}{\int }_a^{b}f\left(t\right)\mathrm{d}t∥}_{[a,b],p}$$ and the $\Delta$-seminorms $${\parallel f\parallel }_p^{\Delta }:={\left({\int }_a^b{\int }_a^b{|f\left(t\right)-f\left(s\right)|}^p\mathrm{d}t\mathrm{d}s\right)}^{\frac{1}{p}}$$ are established. Applications for mid-point and trapezoid inequalities are provided as well.

Let $f$ be a continuous function on $I$ and $A$, $B\in {\mathrm{𝒮𝒜}}_I\left(H\right)$, the convex set of selfadjoint operators with spectra in $I$. If $A\ne B$ and $f$, as an operator function, is Gateaux differentiable on $$[A,B]:=\left\{(1-t)A+tB\mid t\in \left[0,1\right]\right\},$$ while $p:\left[0,1\right]\to ℝ$ is Lebesgue integrable, then we have the inequalities $$\begin{array}{cc}\hfill \parallel {\int }_0^{1}p\left(\tau \right)& f\left(\left(1-\tau \right)A+\tau B\right)d\tau -{\int }_0^{1}p\left(\tau \right)d\tau {\int }_0^{1}f\left(\left(1-\tau \right)A+\tau B\right)d\tau \parallel \hfill \\ & \le {\int }_0^1\tau (1-\tau )|\frac{{\int }_{\tau }^{1}p\left(s\right)ds}{1-\tau }-\frac{{\int }_0^{\tau }p\left(s\right)ds}{\tau }|∥\nabla f_{\left(1-\tau \right)A+\tau B}\left(B-A\right)∥d\tau \hfill \\ & \le \frac{1}{4}{\int }_0^1|\frac{{\int }_{\tau }^{1}p\left(s\right)ds}{1-\tau }-\frac{{\int }_0^{\tau }p\left(s\right)ds}{\tau }|∥\nabla f_{\left(1-\tau \right)A+\tau B}\left(B-A\right)∥d\tau ,\hfill \end{array}$$ where $\nabla f$ is the Gateaux derivative of $f$.

In this paper we point out an Ostrowski type inequality for convex functions which complement in a sense the recent results for functions of bounded variation and absolutely continuous functions. Applications in connection with the Hermite-Hadamard inequality are also considered.

Some trace inequalities of Shisha-Mond type for operators in Hilbert spaces are provided. Applications in connection to Grüss inequality and for convex functions of selfadjoint operators are also given.

Let [...] f(z)=∑n=0∞αnzn $f\left(z\right)={\sum }_{n=0}^{\infty }{\alpha }_n{z}^n$ be a function defined by power series with complex coefficients and convergent on the open disk D (0, R) ⊂ ℂ, R > 0. For any x, y ∈ ℬ, a Banach algebra, with ‖x‖, ‖y‖ < R we show among others that [...] ‖f(y)−f(x)‖≤‖y−x‖∫01fa′(‖(1−t)x+ty‖)dt $$∥f\left(y\right)-f\left(x\right)∥\le ∥y-x∥{\int }_0^1{f}_a^{\text{'}}\left(∥(1-t)x+ty∥\right)dt$$ where [...] fa(z)=∑n=0∞|αn| zn $f_a\left(z\right)={\sum }_{n=0}^{\infty }|{\alpha }_n|z^n$ . Inequalities for the commutator such as [...] ‖f(x)f(y)−f(y)f(x)‖≤2fa(M)fa′(M)‖y−x‖, $$∥f\left(x\right)f\left(y\right)-f\left(y\right)f\left(x\right)∥\le 2f_a\left(M\right)f_a^{\text{'}}\left(M\right)∥y-x∥,$$ if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.

Some reverse Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces are provided. Applications for some convex functions of interest and reverses of Hölder and Schwarz trace inequalities are also given.

Some new inequalities of Hermite-Hadamard type for GA-convex functions defined on positive intervals are given. Refinements and weighted version of known inequalities are provided. Some applications for special means are also obtained.

Some operator inequalities for convex functions of commuting selfadjoint operators that are related to the Hermite-Hadamard inequality are given. Natural examples for some fundamental convex functions are presented as well.