Matrix Inequalities for Convex Functions

AbstractMany converses of Jensen's inequality for convex functions can be found in the literature. Here we give matrix versions, with matrix weights, of these inequalities. Some applications to the Hadamard product of matrices are also given

On Kedlaya type inequalities for weighted means

In 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean $\mathscr{M}$ the Kedlaya-type inequality $$\mathscr{A}\big(x_1,\mathscr{M}(x_1,x_2),\ldots,\mathscr{M}(x_1,\ldots,x_n)\big)\le \mathscr{M} \big(x_1, \mathscr{A}(x_1,x_2),\ldots,\mathscr{A}(x_1,\ldots,x_n)\big)$$ holds for an arbitrary $(x_n)$ ($\mathscr{A}$ stands for the arithmetic mean). We are going to prove the weighted counterpart of this inequality. More precisely, if $(x_n)$ is a vector with corresponding (non-normalized) weights $(\lambda_n)$ and $\mathscr{M}_{i=1}^n(x_i,\lambda_i)$ denotes the weighted mean then, under analogous conditions on $\mathscr{M}$, the inequality $$\mathscr{A}_{i=1}^n \big(\mathscr{M}_{j=1}^i (x_j,\lambda_j),\:\lambda_i\big) \le \mathscr{M}_{i=1}^n \big(\mathscr{A}_{j=1}^i (x_j,\lambda_j),\:\lambda_i\big)$$ holds for every $(x_n)$ and $(\lambda_n)$ such that the sequence $(\frac{\lambda_k}{\lambda_1+\cdots+\lambda_k})$ is decreasing.Comment: J. Inequal. Appl. (2018

About the Neuberg-Pedoe and the Oppenheim inequalities

AbstractThis paper is a complete review and unified treatment of recent results conerning the Neuberg-Pedoe and Oppenheim inequalities. Some new proofs and generalizations of these results are also added

Weighted averages of n-convex functions via extension of Montgomery’s identity

Using an extension of Montgomery’s identity and the Green’s function, we obtain new identities and related inequalities for weighted averages of n-convex functions, i.e. the sum ∑i=1mρih(λi) and the integral ∫abρ(λ)h(γ(λ))dλ where h is an n-convex function. © 2019, The Author(s)

Positivity of sums and integrals for n-convex functions via the Fink identity

We consider the positivity of the sum Σ i=1 ; n ρ i F(ξ i ), where F is a convex function of higher order, as well as analogous results involving the integral ∫ a0 b0 ρ(ξ)F(g(ξ))dξ . We use a representation of the function F via the Fink identity and the Green function that leads us to identities from which we obtain conditions for positivity of the above-mentioned sum and integral. We also obtain bounds for the integral remainders which occur in these identities, as well as corresponding mean value results. © Tübitak