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Authors: Khan, Shahid
Keywords: Mathematics
Issue Date: 2022
Publisher: University of Peshawar, Peshawar
Abstract: This thesis proposes new estimates for the Jensen gap for various classes of functions. These estimates are presented in the form of certain bounds, converses and improvements. In the rst chapter, mathematical inequalities with convexity and their generalizations are discussed. The Jensen inequality and some other classical inequalities with some means are presented. Csisz ar divergence with its special cases and Zipf-Mandelbrot entropy are also given. Second chapter presents some preliminary results such as Green functions with their related identities and some basic lemmas. The third and the fourth chapter present some bounds for the Jensen gap in discrete as well as in integral form for rst and twice di erentiable functions respectively. Consequently some variants of the H older inequality, bounds for the Hermite-Hadamard and Jensen-Ste ensen gaps and inequalities for the power and quasi-arithmetic means are deduced. Similarly, some estimates for the Csisz ar divergence and Zipf-Mandelbrot entropy are obtained. Various proposed results around the Jensen gap are also demonstrated through numerical experiments. In the fth chapter, some new improvements are demonstrated in discrete as well as in integral versions. The aforesaid results enable to give some improvements of the Jensen-Ste ensen, Hermite-Hadamard and H older inequalities. The improved Jensen inequality enables to give, new bounds for the geometric, power, and quasi-arithmetic means. Finally, estimates for various divergences and for the Zipf-Mandelbrot entropy are presented. Similarly the sixth chapter presents some new converses of the Jensen inequality in discrete as well as in integral form. Consequently, new variants of the H older, Hermite-Hadamard and Levinson inequalities are deduced. Some estimates for various divergences in information theory are also established. In the seventh chapter, bounds for the Jensen gap are established in discrete as well as in integral form for s-convex and quasi-convex functions with numerical examples. Consequently, some converses of the H older inequality, bounds for the Hermite-Hadamard gap and estimates for various divergences are obtained. Summary and concluding remarks are given in the nal chapter.
Gov't Doc #: 26072
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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