On Hilbert type inequalities

Abstract

In the present paper we establish new inequalities similar to the extensions of Hilbert's double-series inequality and also give their integral analogues. Our results provide some new estimates to these types of inequalities. MSC: 26D15

Some new discrete Hilbert’s inequalities involving Fenchel–Legendre transform

“…The constant π cosec(π/p) in (1) and (2) is optimal, see [1]. Inequalities (1) and (2) have many generalizations, see for instance [2][3][4] and the references therein, these refinements and ameliorations of the original inequality lead to an important development and improvement of many advanced mathematical branches, see for example [5][6][7].…”

On the Validity of Nonlinear and Nonsmooth Inequalities

“…Several generalizat ions and proofs are given related to the inequalities, such as Jensen type inequalities [1][2][3][4], Hermite-Hadamard inequality [5][6][7][8][9][10], etc. But, it should always be remembered that there is no standard way of proof and there is no general rule in choosing techniques to be used.…”