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We prove an operator arithmetic-harmonic mean type inequality in Krein space setting, by using some block matrix techniques of indefinite type. We also give an example which shows that the operator arithmetic-geometric-harmonic mean inequality for two invertible selfadjoint operators on Krein spaces is not valid, in general.

In this paper, we aim to  state well-known Ostrowski inequality via fractional Montgomery identity for the class of $\phi-\lambda-$ convex functions. This generalized class of convex function contains other well-known convex functions from literature, allowing us to derive Ostrowski-type inequalities as specific instances. Moreover, we present Ostrowski-type inequalities for which certain Powers of absolute derivatives are $\phi-\lambda-$ convex using various techniques, including Hölder's inequality and the Power mean inequality. Consequently, various established results would be captured as special cases. Moreover, we provide applications in terms of special means, allowing us to derive many numerical inequalities related to special means from Ostrowski-type inequalities.

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In this paper, we are introducing for the first time a generalized class named the class of $(\alpha,\beta,\gamma,\delta)-$convex functions of mixed kind. This generalized class contains many subclasses including the class of $(\alpha,\beta)-$convex functions of the first and second kind, $(s,r)-$convex functions of mixed kind, $s-$convex functions of the first and second kind, $P-$convex functions, quasi-convex functions and the class of ordinary convex functions. In addition, we would like to state the generalization of the classical Ostrowski inequality via fractional integrals, which is obtained for functions whose first derivative in absolute values is $(\alpha,\beta,\gamma,\delta)-$ convex function of mixed kind. Moreover, we establish some Ostrowski-type inequalities via fractional integrals and their particular cases for the class of functions whose absolute values at certain Powers of derivatives are $(\alpha,\beta,\gamma,\delta)-$ convex functions of mixed kind using different techniques including H\"older's inequality and Power mean inequality. Also, various established results would be captured as special cases. Moreover, the applications of special means will also be discussed.

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‎Suppose that $B_1,\cdots,B_m$ are positive operators on a Hilbert space $\mathcal{H}$‎. ‎In this paper we generalize the weighted arithmetic‎, ‎geometric and harmonic means as follows‎:‎\begin{align*}‎‎{\mathbf a_m}(\boldsymbol\kappa;\mathbf{B})&={\mathbf a_2}(k_1,N';B_1,{\mathbf a_{m-1}}(\boldsymbol\kappa';\mathbf{B}'))=\frac{k_1B_1+\cdots+k_mB_m}{N}\\‎‎{\mathbf h_m}(\boldsymbol\kappa;\mathbf{B})&={\mathbf h_2}(k_1,N';B_1,{\mathbf h_{m-1}}(\boldsymbol\kappa';\mathbf{B}'))=\left(\frac{k_1B_1^{-1}+\cdots+k_mB_m^{-1}}{N}\right)^{-1}\\‎‎{\mathbf g_m}(\boldsymbol\kappa;\mathbf{B})&={\mathbf g_2}(k_1,N';B_1,{\mathbf g_{m-1}}(\boldsymbol\kappa';\mathbf{B}'))‎‎\end{align*}‎‎where $\boldsymbol\kappa=(k_1,\cdots,k_m)‎, ‎N=k_1+\cdots+k_m‎, ‎\boldsymbol\kappa'=(k_2,\cdots‎, ‎k_m)$ and $N'=k_2+\cdots+k_m$‎. ‎We show that the arithmetic-geometric-harmonic mean inequality holds‎. ‎Also we investigate nine property of the geometric mean‎.

In this paper, we establish some Hermite{Hadamard type inequalities for function whose n-th derivatives are logarithmically convex by using Riemann{Liouville integral operator.

In this paper, the generalized a-centroidal mean and its dual form in 2 variables are introduced. Also, we will study some properties and prove their monotonicity. Further, shown that various means are particular cases of generalized a-centroidal mean.