commun. fac. sci. univ. ank. ser. a1 math. stat. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 1303-5991 2618-6470 Ankara University 10.31801/cfsuasmas.1262668 Mathematical Sciences Matematik New proofs of Fejer's and discrete Hermite-Hadamard inequalities with applications https://orcid.org/0000-0001-7176-5164 Sekin Çağla AKDENIZ UNIVERSITY https://orcid.org/0000-0001-8933-8769 Tamar Mehmet Emin ABDULLAH GUL UNIVERSITY https://orcid.org/0000-0003-2353-7700 Aliyev İlham AKDENIZ UNIVERSITY 12 29 2023 72 4 1110 1125 03 09 2023 06 22 2023 Copyright © 1948, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 1948 Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics

New proofs of the classical Fejer inequality and discrete Hermite-Hadamard inequality (HH) are presented and several applications are given, including (HH)-type inequalities for the functions, whose derivatives have inflection points. Morever, some estimates from below and above for the first moments of functions $f:[a,b]\rightarrow \mathbb{R}$ about the midpoint $c=(a+b)/2$ are obtained and the reverse Hardy inequality for convex functions $f:(0,\infty )\rightarrow (0,\infty )$ is established.

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