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Polynomial selections and separation by polynomials

100%

Wąsowicz S.

Studia Mathematica

1996

tom 120

nr 1

75-82

K. Nikodem and the present author proved in [3] a theorem concerning separation by affine functions. Our purpose is to generalize that result for polynomials. As a consequence we obtain two theorems on separation of an n-convex function from an n-concave function by a polynomial of degree at most n and a stability result of Hyers-Ulam type for polynomials.

New bounds for Shannon, Relative and Mandelbrot entropies via Hermite interpolating polynomial

80%

Mehmood N. , Butt S. I. , Pečarić D. , Pečarić J.

Demonstratio Mathematica

2018

tom Vol. 51, nr 1

112--130

To procure inequalities for divergences between probability distributions, Jensen’s inequality is the key to success. Shannon, Relative and Zipf-Mandelbrot entropies have many applications in many applied sciences, such as, in information theory, biology and economics, etc. We consider discrete and continuous cyclic refinements of Jensen’s inequality and extend them from convex function to higher order convex function by means of different new Green functions by employing Hermite interpolating polynomial whose error term is approximated by Peano’s kernal. As an application of our obtained results, we give new bounds for Shannon, Relative and Zipf-Mandelbrot entropies.