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Physical Sciences and Mathematics Commons

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Mathematics

Turkish Journal of Mathematics

Journal

2017

Convex function

Articles 1 - 2 of 2

Full-Text Articles in Physical Sciences and Mathematics

More Accurate Jensen-Type Inequalities For Signed Measures Characterized Via Green Function And Applications, Mario Krnic, Josip Pecaric, Mirna Rodic Jan 2017

More Accurate Jensen-Type Inequalities For Signed Measures Characterized Via Green Function And Applications, Mario Krnic, Josip Pecaric, Mirna Rodic

Turkish Journal of Mathematics

In this paper we derive several improved forms of the Jensen inequality, giving the necessary and sufficient conditions for them to hold in the case of the real Stieltjes measure not necessarily positive.The obtained relations are characterized via the Green function. As an application, our main results are employed for constructing some classes of exponentially convex functions and some Cauchy-type means.


Asymptotic For A Second-Order Evolution Equation With Convex Potential Andvanishing Damping Term, Ramzi May Jan 2017

Asymptotic For A Second-Order Evolution Equation With Convex Potential Andvanishing Damping Term, Ramzi May

Turkish Journal of Mathematics

In this short note, we recover by a different method the new result due to Attouch, Chbani, Peyrouqet, and Redont concerning the weak convergence as $t\rightarrow+\infty$ of solutions $x(t)$ to the second-order differential equation $x^{\prime\prime}(t)+\frac{K}{t}x^{\prime}(t)+\nabla\Phi(x(t))=0,$ where $K>3$ and $\Phi$\ is a smooth convex function defined on a Hilbert space $\mathcal{H}.$ Moreover, we improve their result on the rate of convergence of $\Phi(x(t))-\min\Phi.$