We conjecture that the Parisi functional in the Sherrington-Kirkpatrick model is convex in the functional order parameter. We prove a partial result that shows the convexity along “one-sided” directions. An interesting consequence of this result is the log-convexity of Lm norm for a class of random variables.

We establish a positivity property for the difference of products of certain Schur functions, sλ(x), where λ varies over a fundamental Weyl chamber in R n and x belongs to the positive orthant in R. Further, we generalize that result to the difference of certain products of arbitrary numbers of Schur functions. We also derive a log-convexity property of the generalized hypergeometric functions ...

We prove a conjecture of Liu and Wang on the q-log-convexity of the Narayana polynomials of type B. By using Pieri’s rule and the JacobiTrudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. By the principal specialization this, leads to q-logconvexity. We also show that the lin...

It is known that the concept of ratio monotonicity is closely related to log-convexity and log-concavity. In this paper, by exploring the log-behavior properties of a new combinatorial sequence defined by Z.-W. Sun, we completely solve a conjecture on ratio monotonicity by him.

We continue the investigation of computational aspects of restricted-orientation convexity (O-convexity) in two dimensions. We introduce one notion of an O-halfplane, for a set O of orientations, and we investigate O-connected convexity. The O-connected convex hull of a nite set X can be computed in time O(jXj log jXj + jOj). The O-connected hull is a basis for determining the O-convex hull of ...

This paper considers the generalized Stirling numbers of the first and second kinds. First, we show that the sequences of the above generalized Stirling numbers are both log-concave under some mild conditions. Then, we show that some polynomials related to the above generalized Stirling numbers are q-log-concave or q-log-convex under suitable conditions. We further discuss the log-convexity of ...

In this article the invariant centroidal mean is introduced. A pair of double sequences in term Centroidal and its are defined discussed properties monotonicity log-convexity log-concavity. Finally as an illustration it justified that new Gaussian compound CT' ⊗ CT converging faster than H A. 2000 Mathematics Subject Classification. Primary 26D15.