Volume 32, Issue 4 -

In the paper, the authors discuss the Schur-harmonic convexity on (0,∞) × (0,∞) for differences of some famous means, such as the arithmetic, geometric, harmonic, root-square means, and the like, and obtain some inequalities related to differences of means.

It is proved that the maximal operator of the Weierstrass and Picard summability means of a tempered distribution is bounded from H p (ℝ d ) to L p (ℝ d ) for all 0 <  p  ≤ ∞ and, consequently, is of weak type (1,1). As a consequence we obtain that the summability means of a function f  ∈  L 1 (ℝ d ) converge a.e. to f . Similar results are shown for conjugate functions and for Fourier series.

The advantages of framelet packets and their promising features in various applications have attracted a lot of interest and effort in recent years. In this paper, we present an explicit construction of M -band tight framelet packets via unitary extension principle.

We prove that weak solutions of uniformly elliptic equations in divergence form in the plane posses the strong unique continuation property.

We obtain a variety of series and integral representations of the digamma function ψ ( a ). These in turn provide representations of the evaluations ψ ( p/q ) at rational argument and for the polygamma function ψ ( j ) . The approach is through a limit definition of the zeroth Stieltjes constant γ 0 ( a ) = - ψ ( a ). Several other results are obtained, including product representations for exp[ γ 0 ( a )] and for the Gamma function Γ( a ). In addition, we present series representations in terms of trigonometric integrals Ci and Si for ψ ( a ) and the Euler constant γ  = - ψ (1).