New sharp inequalities of Ostrowski and generalized trapezoid type for the Riemann-Stieltjes integrals and applications

In this paper, new sharp weighted generalizations of Ostrowski and generalized trapezoid type inequalities for the Riemann--Stieltjes integrals are proved. Several related inequalities are deduced and investigated. New Simpson's type inequalities for $\mathcal{RS}$--integral are pointed out. Finally, as application; an error estimation of a general quadrature rule for $\mathcal{RS}$--integral via Ostrowski--generalized trapezoid quadrature formula is given.Comment: 22 page

q-Bernoulli Inequality

In this work, the q-analogue of Bernoulli inequality is proved. Some other related results are presented.Comment: 7 page

On Pompeiu-Chebyshev functional and its generalization

In this work, a generalization of Chebyshev functional is presented. New inequalities of Gruss type via Pompeiu's mean value theorem are established. Improvements of some old inequalities are proved. A generalization of pre-Gruss inequality is elaborated. Some remarks to further generalization of Chebyshev functional are presented. As applications, bounds for the reverse of CBS inequality are deduced. Hardy type inequalities on bounded real interval [a,b] under some other circumstances are introduced. Other related ramified inequalities for differentiable functions are also given.Comment: 30 page

The Hermite-Hadamard inequality on hypercuboid

Given any ${\bf{a}}: = \left( {a_1 ,a_2 , \ldots ,a_n } \right)$ and ${\bf{b}}: = \left( {b_1 ,b_2 , \ldots ,b_n } \right)$ in $\mathbb{R}^n$. The $\textbf{n}$-fold convex function defined on $\left[ {{\bf{a}},{\bf{b}}} \right]$, ${\bf{a}},{\bf{b}} \in \mathbb{R}^n$ with ${\bf{a}}<{\bf{b}}$ is a convex function in each variable separately. In this work we prove an inequality of Hermite-Hadamard type for $\textbf{n}$-fold convex functions. Namely, we establish the inequality \begin{align*} f\left( {\frac{{{\bf{a}} + {\bf{b}}}}{2}} \right) \le \frac{1}{{{\bf{b}} - {\bf{a}}}}\int_{\bf{a}}^{\bf{b}} {f\left( {\bf{x}} \right)d{\bf{x}}} \le \frac{1}{{2^n }}\sum\limits_{\bf{c}} {f\left( {\bf{c}} \right)}, \end{align*} where $\sum\limits_{\bf{c}} {f\left( {\bf{c}} \right)} : = \sum\limits_{\mathop {c_i \in \left\{ {a_i ,b_i } \right\}}\limits_{1 \le i \le n} } {f\left( {c_1, c_2, \ldots ,c_n } \right)}$. Some other related result are given.Comment: 12 page

Pompeiu-Chebyshev type inequalities for selfadjoint operators in Hilbert spaces

In this work, generalization of some inequalities for continuous $h$-synchronous ($h$-asynchronous) functions of selfadjoint linear operators in Hilbert spaces are proved.Comment: 14 page

On Alzer's inequality

Extensions and generalizations of Alzer's inequality; which is of Wirtinger type are proved. As applications, sharp trapezoid type inequality and sharp bound for the geometric mean are deduced.Comment: 8 page

An Inequality of Simpson's type Via Quasi-Convex Mappings with Applications

In this paper, an inequality of Simpson type for quasi-convex mappings are proved. The constant in the classical Simpson's inequality is improved. Furthermore, the obtained bounds can be (much) better than some recently obtained bounds. Application to Simpson's quadrature rule is also given.Comment: 7 pages, no figur

Popoviciu's type inequalityies for h-MN-convex functions

In this work, several inequalities of Popoviciu type for h-MN-convex functions are proved, where M or N are denote to Arithmetic, Geometric and Harmonic means and $h$ is a non-negative superadditive or subadditive function.Comment: 26 page

New Inequalities of Steffensen's type for s-convex functions

In this work, new inequalities connected with the Steffensen's integral inequality for s-convex functions are provedComment: 8 page

Some properties of h-MN-convexity and Jensen's type inequalities

In this work, we introduce the class of $h$-${\rm{MN}}$-convex functions by generalizing the concept of ${\rm{MN}}$-convexity and combining it with $h$-convexity. Namely, Let $I,J$ be two intervals subset of $\left(0,\infty\right)$ such that $\left(0,1\right)\subseteq J$ and $\left[a,b\right]\subseteq I$. Consider a non-negative function $h: (0,\infty)\to \left(0,\infty\right)$ and let ${\rm{M}}:\left[0,1\right]\to \left[a,b\right]$ $(0<a<b)$ be a Mean function given by ${\rm{{\rm{M}}}}\left(t\right)={\rm{{\rm{M}}}}\left( {h(t);a,b} \right)$; where by ${\rm{{\rm{M}}}}\left( {h(t);a,b} \right)$ we mean one of the following functions: $A_h\left( {a,b} \right):=h\left( {1 - t} \right)a + h(t) b$, $G_h\left( {a,b} \right)=a^{h(1-t)} b^{h(t)}$ and $H_h\left( {a,b} \right):=\frac{ab}{h(t) a + h\left( {1 - t} \right)b} = \frac{1}{A_h\left( {\frac{1}{a},\frac{1}{b}} \right)}$; with the property that ${\rm{{\rm{M}}}}\left( {h(0);a,b} \right)=a$ and ${\rm{M}}\left( {h(1);a,b} \right)=b$. A function $f : I \to \left(0,\infty\right)$ is said to be $h$-${\rm{{\rm{MN}}}}$-convex (concave) if the inequality \begin{align*} f \left({\rm{M}}\left(t;x, y\right)\right) \le (\ge) \, {\rm{N}}\left(h(t);f (x), f (y)\right), \end{align*} holds for all $x,y \in I$ and $t\in [0,1]$, where M and N are two mean functions. In this way, nine classes of $h$-${\rm{MN}}$-convex functions are established and some of their analytic properties are explored and investigated. Characterizations of each type are given. Various Jensen's type inequalities and their converses are proved.Comment: 27 pages. Journal of Interdisciplinary Mathematics 201