Positivity of sums and integrals for n-convex functions via the Fink identity and new Green functions

We consider positivity of sum $\sum_{i=1}^np_if(x_i)$ involving convex functions of higher order. Analogous for integral $\int_a^bp(x)f(g(x))dx$ is also given. Representation of a function $f$ via the Fink identity and the Green function leads us to identities for which we obtain conditions for positivity of the mentioned sum and integral. We obtain bounds for integral remainders which occur in those identities as well as corresponding mean value theorems.

On Opial-Type Inequalities for Superquadratic Functions and Applications in Fractional Calculus

We have studied the Opial-type inequalities for superquadratic functions proved for arbitrary kernels. These are estimated by applying mean value theorems. Furthermore, by analyzing specific functions, the fractional integral and fractional derivative inequalities are obtained.

Generalized Conformable Mean Value Theorems with Applications to Multivariable Calculus

The conformable derivative and its properties have been recently introduced. In this research work, we propose and prove some new results on the conformable calculus. By using the definitions and results on conformable derivatives of higher order, we generalize the theorems of the mean value which follow the same argument as in the classical calculus. The value of conformable Taylor remainder is obtained through the generalized conformable theorem of the mean value. Finally, we introduce the conformable version of two interesting results of classical multivariable calculus via the conformable formula of finite increments.

Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals

The cumulative distribution function of the non-central chi-square distribution χn′2(λ) of n degrees of freedom possesses an integral representation. Here we rewrite this integral in terms of a lower incomplete gamma function applying two of the second mean-value theorems for definite integrals, which are of Bonnet type and Okamura’s variant of the du Bois–Reymond theorem. Related results are exposed concerning the small argument cases in cumulative distribution function (CDF) and their asymptotic behavior near the origin.

Hardy-Type Inequalities for an Extension of the Riemann- Liouville Fractional Derivative Operators

In this paper we present variety of Hardy-type inequalities and their refinements for an extension of Riemann-Liouville fractional derivative operators. Moreover, we use an extension of extended Riemann-Liouville fractional derivative and modified extension of Riemann-Liouville fractional derivative using convex and monotone convex functions. Furthermore, mean value theorems and n-exponential convexity of the related functionals is discussed.

Mean value theorems for polynomial solutions of linear elliptic equations with constant coefficients in the complex plane

We characterize solutions of the mean value linear elliptic equation with constant coefficients in the complex plane in the case of regular polygon.