Hadamard, Jensen and related inequalities

 

Summary

 

The thesis explores integral inequalities for convex functions of higher order and integral inequalities involving higher order bound derivatives. The guiding line present Hadamard, Jensen and related inequalities (either generalized or used in the proofs of new inequalities).

The main results relate to the generalization or the improvement of known inequalities:

Hadamard's, conversion of Jensen's inequality, Petrović's, Iyengar's, Wirtinger's and Kesava Menon's inequalities.

Generalizations of Hadamard's inequalities and the conversion of Jensen's inequality for (n)-convex function and with conditions on the real Borel measure are presented by Lidstone's and Hermite's interpolating polynomials. As a discrete form we also obtain a generalization of Petrovć's inequality, i.e. Giaccardi's inequality.

Apart from the extension (weighted and integral form) of the more recent generalization of discrete Wirtinger's inequality, the thesis also explains the improvement of these results by considerably relaxing the condition on functions involved. The connection between Wirtinger's and Jensen's inequality has been observed.

Some extensions of Iyengar's inequality for functions of bounded n-th derivative, depending on evenness n, are obtained by the use of Hayashi's modification of Steffensen's inequality.

Kesava Menon inequalities are connected to the tails of Taylor's series. The original statements are improved, under the same or weaker conditions on functions involved. The thesis also refers to the error when comparing Menon's, Iyengar's and Hadamard's inequalities.

Further, using Jensen's and Jensen-Steffensen's inequality, thesis establish the right side of the Hadamard inequality for log-concave functions in case of the positive measure and in case of the real Borel measure along with additional conditions.

An inequality associated with Lidstone polynomials, for nonnegative, concave function with bounded 2n-th derivatives is also improved. An upper bound for the error function of the Hermite interpolating polynomial ||eH(x)|| in terms of ||f(n)||n is given and then used to obtain various generalizations of Mahajani's inequality for a different norm n .