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On co-ordinated quasi-convex functions

M. Emin Özdemir, Ahmet Ocak Akdemir, Çetin Yıldız (2012)

Czechoslovak Mathematical Journal

A function f : I , where I is an interval, is said to be a convex function on I if f ( t x + ( 1 - t ) y ) t f ( x ) + ( 1 - t ) f ( y ) holds for all x , y I and t [ 0 , 1 ] . There are several papers in the literature which discuss properties of convexity and contain integral inequalities. Furthermore, new classes of convex functions have been introduced in order to generalize the results and to obtain new estimations. We define some new classes of convex functions that we name quasi-convex, Jensen-convex, Wright-convex, Jensen-quasi-convex and Wright-quasi-convex functions...

On equivalence of super log Sobolev and Nash type inequalities

Marco Biroli, Patrick Maheux (2014)

Colloquium Mathematicae

We prove the equivalence of Nash type and super log Sobolev inequalities for Dirichlet forms. We also show that both inequalities are equivalent to Orlicz-Sobolev type inequalities. No ultracontractivity of the semigroup is assumed. It is known that there is no equivalence between super log Sobolev or Nash type inequalities and ultracontractivity. We discuss Davies-Simon's counterexample as the borderline case of this equivalence and related open problems.

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