convex functions

This study investigates the geometric characteristics of integral operators connected with the Pascal distribution series. The objective of this article is to establish conditions on the parameters of these operators by utilizing coefficient bounds from class $\mathcal{R}_{\wp,\upsilon}^{\tau}(\gamma)$. Also, the study analyzes the inclusion outcomes of integral operators related to the Pascal distribution series within various subclasses of analytic functions. The findings presented here incorporated previously published results as specific instances.

In this paper, we investigate the Bullen inequality in the context of fractional integrals with exponential kernels. Building upon the foundational works in the field, we first introduce a new integral identity. From this identity, we derive several novel Bullen-type inequalities for differentiable convex functions. To validate our theoretical findings, we provide a numerical example along with graphical representations, demonstrating the accuracy and applicability of our results. The results obtained are not only new for the fractional integrals considered in our study, but as \(\alpha\) approaches 1, we also derive additional novel results for the classical integral.

In this paper, we introduce the subclass $\mathcal{K}\mathcal{S}(\alpha)$ of univalent functions in $\mathcal{A}$ and study some properties of this class. We apply matters of differential subordinations, to investigate some results concerning the subclasses $\mathcal{K}\mathcal{S}(\alpha)$ and $\mathcal{B}\mathcal{S}(\alpha)$ of $\mathcal{A}$, where $\alpha \in [0,1)$.

The main objective of this paper is to establish some new inequalities related to the open Newton-Cotes formulas in the setting of q-calculus. We establish a quantum integral identity first and then prove the desired inequalities for $q$-differentiable convex functions. These inequalities are useful for determining error bounds for the open Newton-Cotes formulas in both classical and $q$-calculus. This work distinguishes itself from existing studies by employing quantum operators, leading to sharper and more precise error estimates. These results extend the applicability of Newton-Cotes methods to quantum calculus, offering a novel contribution to the numerical analysis of convex functions. Finally, we provide mathematical examples and computational analysis to validate the newly established inequalities.

In this investigation, using Opoola differential operator ($D^{m}(\mu,\beta,t)f(z)$), a new integral operator: $I_{t,\beta,\mu}^{m,\sigma}(f_{1},...,f_{n})(z): A^{n}\rightarrow A$  is defined in the unit disk, $U=\left\lbrace z\in C:\left|z\right|<1\right\rbrace$; and we investigated the Univalence conditions of this generalized operator. Finally, a number of corollaries and remarks which show the extension of our results are presented.

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.

This work introduces the quantum analogue of the dual Simpson type integral inequalities for the class of q-differentiable convex functions through a new identity. The results are also accompanied by their applications.

In this paper we introduce a new sequence of mappings in connection to Hermite-Hadamard type inequality. Some bounds and refinements of Hermite-Hadamard inequality for convex functions via this  sequence  are given

The purpose of this article is to make a connection between the Pascal distribution series and some subclasses of normalized analytic functions whose coefficients are probabilities of the Pascal distribution. To be more precise, we investigate such connections with the classes of parabolic starlike and uniformly convex functions with positive coefficients in the open unit disk $\mathbb{U}.$

In this article, some integral equations are obtained and based on these integral equations, new integral inequalities are obtained for convex functions that satisfy certain convexity conditions.

We first construct new Hermite-Hadamard type inequalities which include generalized fractional integrals for convex functions by using an operator which generates some significant fractional integrals such as Riemann-Liouville fractional and the Hadamard fractional integrals. Afterwards, Trapezoid and Midpoint type results involving generalized fractional integrals for functions whose the derivatives in modulus and their certain powers are convex are established. We also recapture the previous results in the particular situations of the inequalities which are given in the earlier works.

In this paper, several novel inequalities are examined for the product of two s-convex functions in the fourth sense. Also, some applications regarding special means and digamma functions are presented.

Let $f$ be a convex function on $I$ and $a,$ $bin I$ with $a<b.$ If $p:% left[ a,bright] rightarrow lbrack 0,infty )$ is Lebesgue integrable and symmetric, namely $pleft( b+a-tright) =pleft( tright) $ for all $tin % left[ a,bright] ,$ then we show in this paper that begin{align*} 0& leq frac{1}{2}int_{a}^{b}leftvert t-frac{a+b}{2}rightvert pleft( tright) dtleft[ f_{+}^{prime }left( frac{a+b}{2}right) -f_{-}^{prime }left( frac{a+b}{2}right) right]  \ & leq int_{a}^{b}pleft( tright) fleft( tright) dt-left( int_{a}^{b}pleft( tright) dtright) fleft( frac{a+b}{2}right)  \ & leq frac{1}{2}int_{a}^{b}leftvert t-frac{a+b}{2}rightvert pleft( tright) dtleft[ f_{-}^{prime }left( bright) -f_{+}^{prime }left( aright) right] end{align*} and begin{align*} 0& leq frac{1}{2}int_{a}^{b}left[ frac{1}{2}left( b-aright) -leftvert t-frac{a+b}{2}rightvert right] pleft( tright) dtleft[ f_{+}^{prime }left( frac{a+b}{2}right) -f_{-}^{prime }left( frac{a+b}{% 2}right) right]  \ & leq left( int_{a}^{b}pleft( tright) dtright) frac{fleft( aright) +fleft( bright) }{2}-int_{a}^{b}pleft( tright) fleft( tright) dt \ & leq frac{1}{2}int_{a}^{b}left[ frac{1}{2}left( b-aright) -leftvert t-frac{a+b}{2}rightvert right] pleft( tright) dtleft[ f_{-}^{prime }left( bright) -f_{+}^{prime }left( aright) right] . end{align*}.

We extend the definitions of $nabla-$convex and completely monotonic functions for two variables. Some general identities of Popoviciu type integrals $int P(y)f(y) dy$ and $int int P(y,z) f(y,z) dy   dz$ are deduced. Using obtained identities, positivity of these expressions are characterized for  higher order $nabla-$convex and completely monotonic functions. Some applications in terms of generalized Cauchy means and exponential convexity are given.

In this article, we introduce the notion of $(h,k)$-convex functions and their operator form. Moreover, we derive Hermite–Hadamard-type, and Fejer-type inequalities for this class.

The purpose of this paper is to define a new class of analytic, normalized functions in the open unit disk D = {z : z ∈ C and |z| < 1} subordinating with crescent shaped regions, and to derive certain coefficient estimates a2 , a3 and Fekete-Szeg¨o inequality for f ∈ Mq(α, β, λ). A similar result have been done for the function f −1 . Further application of our results to certain functions defined by convolution products with a normalized analytic function is given, in particular we obtain FeketeSzeg¨o inequalities for certai

‎In this paper‎, ‎the authors establish inequalities of the‎ Hermite-Hadamard-Mercer type for convex functions by applying k-fractional‎ integrals‎. ‎We prove some new fractional inequalities connected to the left‎ part of Hermite-Hadamard-Mercer type inequalities for differentiable‎ ‎mappings whose first derivatives in absolute value are convex‎.

In the present paper, we introduce and investigate three interesting superclasses SD, SD* and KD of analytic, normalized and univalent functions in the open unit disk D. For functions belonging to these classes  SD, SD* and  KD, we derive several properties including (for example) the coefficient bounds and growth theorems. The various results presented here would generalize many well known results. We also get a new univalent criterion and some interesting properties for univalent function,starlike function,convex function and close-to-convex function. Many superclasses which are already studied by various researchers are obtained as special cases of our two new superclasses.