A sequence of mappings associated with the Hermite-Hadamard inequalities and applications

summary:New properties for some sequences of functions defined by multiple integrals associated with the Hermite-Hadamard integral inequality for convex functions and some applications are given

On Some Hadamard-Type Inequalıtıes for (r,m) -Convex Functıons

In this paper, we define a new class of convex functions which is called (r,m) - convex functions. We also prove some Hadamard\u27s type inequalities based on this new definition

Design of an essentially non-oscillatory reconstruction procedure in finite-element type meshes

An essentially non oscillatory reconstruction for functions defined on finite element type meshes is designed. Two related problems are studied: the interpolation of possibly unsmooth multivariate functions on arbitary meshes and the reconstruction of a function from its averages in the control volumes surrounding the nodes of the mesh. Concerning the first problem, the behavior of the highest coefficients of two polynomial interpolations of a function that may admit discontinuities of locally regular curves is studied: the Lagrange interpolation and an approximation such that the mean of the polynomial on any control volume is equal to that of the function to be approximated. This enables the best stencil for the approximation to be chosen. The choice of the smallest possible number of stencils is addressed. Concerning the reconstruction problem, two methods were studied: one based on an adaptation of the so called reconstruction via deconvolution method to irregular meshes and one that lies on the approximation on the mean as defined above. The first method is conservative up to a quadrature formula and the second one is exactly conservative. The two methods have the expected order of accuracy, but the second one is much less expensive than the first one. Some numerical examples are given which demonstrate the efficiency of the reconstruction

Integral Inequalities of Hermite-Hadamard Type Via Green Function and Applications

In this study, we establish some Hermite- Hadamard type inequalities for functions whose second derivatives absolute value are convex. In accordance with this purpose, we obtain an identity using Green\u27s function. Then using this equality we get our main results

Second variation techniques for stability in geometric inequalities

We study stability of minimizers for several geometric problems. Applying second variation techniques and some free boundary regularity results we are able to prove sharp quantitative isocapacitary inequality, both in the case of standard capacity and that of p-capacity. With the same approach we deduce that charged liquid droplets minimizing Debye-H\ufcckel-type free energy are spherical in the small charge regime

Interpolation between functions of means

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