Narrow Results

We prove that an equation holds for differentiable convex functions, and this result has been derived using conformable integrals. With the help of this equality, several parameterized inequalities are established by using the conformable fractional integrals. Namely, we show that our main inequalities reduce to Ostrowski-, Hermite–Hadamard-, Simpson-, and Bullen-type inequalities which are proved in earlier published papers. More precisely, some inequalities are acquired by taking advantage of the convexity, the Hölder, and the power mean inequalities. Finally, examples are given to illustrate the investigated results.

The Jensen, Jensen–Mercer, and Hermite–Hadamard inequalities have been key in advancing convex inequalities for scalars and operators, with several applications in fields like mathematical physics, operator theory and mathematical inequalities. This paper presents a new argument leading to interesting refinements of these celebrated inequalities. Applications that include entropy and operator inequalities will be presented.

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if is a polynomial of degree m with non-negative coefficients, then, for all positive operators A, B and all symmetric norms,

The purpose of this paper is to provide novel estimates of Ostrowski-type inequalities in a much simpler and shorter way of some recent significant results in the context of a fractal set ${ℝ}^{\tilde{\alpha }}.$ By using our new approach, we established an auxiliary result that correlates with generalized convex (${𝒢}{𝒞}$) and concave functions for absolutely continuous functions with second-order local differentiable mappings. Moreover, we derived some companions of Ostrowski-type inequalities belonging to ${{𝒱}}^{(2\tilde{\alpha })}\in L_{\infty }[s_1,s_2],{{𝒱}}^{(2\tilde{\alpha })}\in L_p[s_1,s_2]$ and ${{𝒱}}^{(2\tilde{\alpha })}\in L_1[s_1,s_2]$ in local fractional sense. Our results generalize and offer better bounds than many known results in the existing literature associated with trapezoidal and midpoint formula. As an application perspective, we derived several estimation-type outcomes by the use of generalized $\tilde{\alpha }$-type special means formula provided here to illustrate the usability of the obtained results. Our study contributes to a better understanding of fractal analysis and proves beneficial in exploring real-world phenomena.

The purpose of this paper is to study a generalization of strongly $\eta$-convex functions using the local fractional calculus developed by Yang [Advanced Local Fractional Calculus and Its Applications (World Science Publisher, New York, 2012)], namely generalized strongly $\eta$-convex function. Among other results, we obtain some Hermite–Hadamard and Fejér type inequalities for this class of functions.

In this paper, we present new Hermite–Hadamard–Fejér type inequalities via Riemann–Liouville fractional integrals for the function $g(\tau )h(\tau )$, where $g(\tau )$ is convex function and $h(\tau )$ is symmetric with respect to $\frac{a+b}{2}$. Our results in some special cases yield the well-known classic Hermite–Hadamard–Fejér type inequalities. At the end of the paper, the applications of our results are given.

In this research, we study some algebraic and topological properties of generalized $m$-convex functions on fractal sets and establish a new Jensen type inequality. Also, we obtain some new Hermite–Hadamard type inequalities for local fractional integrals of this class of functions. As applications, we present some inequalities with generalized special means and generalized probability density functions.

The following inequality

In this paper, we define parametric $R$-norm directed-divergence convex function and discuss their special cases and prove some properties similar to Kullback–Leibler information measure. From $R$-norm divergence measure new information measures have also been derived and their relations with different measures of entropy have been obtained and give its application in industrial engineering.

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalization of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.

In this paper, we consider generalized solutions to the Dirichlet problem for a class of generalized Monge–Ampère equations. For such generalized solutions, we give a complete proof for the so-called comparison principle.

It is shown that by knowing the slope of a bounded from below lower semicontinuous convex function at each point of a Banach space, we know the function up to an additive constant.

We extend to infinite dimensional separable Hilbert spaces the Schur convexity property of eigenvalues of a symmetric matrix with real entries. Our framework includes both the case of linear, selfadjoint, compact operators, and that of linear selfadjoint operators that can be approximated by operators of finite rank and having a countable family of eigenvalues. The abstract results of the present paper are illustrated by several examples from mechanics or quantum mechanics, including the Sturm–Liouville problem, the Schrödinger equation, and the harmonic oscillator.

This paper presents a geometrìc characterization of convex sets in locally convex spaces on which a strong optimization theorem of the Stegall-type holds, and gives Collier's theorem of ω* Asplund spaces a localized setting.

This paper presents a type of variational principles for real valued w* lower semicontinuous functions on certain subsets in duals of locally convex spaces, and resolve a problem concerning differentiability of convex functions on general Banach spaces. They are done through discussing differentiability of convex functions on nonlinear topological spaces and convexification of nonconvex functions on topological linear spaces.

By making use of the principle of subordination and Dziok–Srivastava operator, we introduce a certain subclass of p-valent non-Bazilevic analytic functions. Such results as subordination and superordination properties, convolution properties, distortion theorems and inequality properties, are proved.

In this paper, we will give some multidimensional generalization of reversed Hardy type inequalities for monotone functions. Moreover, we will give n-exponential convexity, exponential convexity and related results for some functionals obtained from the differences of these inequalities. At the end we will give mean value theorems and Cauchy means for these functionals.

In this paper, we indicate some inequalities for double integrals of functions whose partial derivatives in absolute value are convex on the coordinates on rectangle from the plane. Our established results generalize some recent results for functions whose partial derivatives in absolute value are convex on the coordinates on the rectangle from the plane.

In this paper, we introduce the concept of quasi-convex functions by considering a subclass ${{𝒞}}_p^{\ast }(\lambda ,\beta )$ of normalized analytic functions in the unit disk $𝕌$ for some convex function $g(z)$ with fixed second coefficient of order $\lambda$, such that

We consider discrete and continuous cyclic refinements of Jensen’s inequality and generalize them from convex function to higher order convex function by means of Lagrange Green’s function and Montgomery identity. We give application of our results by formulating the monotonicity of the linear functionals obtained from generalized identities utilizing the theory of inequalities for $n$-convex functions at a point. We compute Grüss and Ostrowski type bounds for generalized identities associated with the obtained inequalities. Finally, we investigate the properties of linear functionals regarding exponential convexity log convexity and mean value theorems.