ABSTRACT. This paper deals with the the norm of the weighted Bergman projection operator Pα : L ∞ (B) → B where α > −1 and B is the Bloch space of the unit ball B of the complex space C n . We consider two Bloch norms, the standard Bloch norm and invariant norm w.r.t. automorphisms of the unit ball. Our work contains as a special case the main result of the recent paper [4].

Abstract. Motivated by some recent results of Kalaj and Vuorinen (Proc. Amer. Math. Soc., 2012), we prove that positive harmonic functions defined in the upper half-plane are contractions w.r.t. hyperbolic metrics of half-plane and positive part of the real line, respectively Introduction and the main resultDenote by U = {z ∈ C : |z| < 1} the unit disc of the complex plane C and by H = {z ∈ C : Im z > 0} the upper half-plane. R is the whole real axis, and the positive real axis is denoted by R + = {x ∈ R : x > 0}.Let d h stands for the hyperbolic distance on the disc U. With the same letter we denote the hyperbolic distance on H and R + , since we believe that misunderstanding will not occur. We havewhere γ ⊆ U is any regular curve connecting z ∈ U and w ∈ U. On the other hand, the hyperbolic distance between z ∈ H and w ∈ H iswhere now γ ⊆ H. In particular, the hyperbolic distance between x ∈ R + and y ∈ R + , where x ≤ y isfor all z, w ∈ U. The equality sign occurs if and only if f is a Möbius transform of U onto itself.The previous result has a counterpart for analytic functions f : H → H. Using the Cayley transform one easily finds that the Schwarz-Pick inequality in this settings says for every z, w ∈ H. Letting z → w in (1), we obtainRegarding the expression for the hyperbolic distance in the upper half-plane, (1) may be rewritten aswhich means that f is a contraction in the hyperbolic metric of H. It is well known that the equality sign attains in (1), (2), and (3) (for some z or for some distinct z and w, and therefore for all such points) if and only if f = a Möbius transform of H onto itself.During the past decade, harmonic mappings and functions have been extensively studied and many results from the theory of analytic functions have been extended for them.Quite recently Kalaj and Vuorinen [3] proved that a harmonic function f : U → (−1, 1) is a Lipschitz function in the hyperbolic metric, i.e., for every z, w ∈ U they obtained that. Actually, using the classical results they firstly establishedBoth inequalities are sharp.We refer to [1] for a related result. We are interested here in the positive harmonic functions defined in H. As we have said in the abstract, our main aim is to prove Theorem 1.1. Let u : H → R + be harmonic. Thenfor all z, w ∈ H. In other words, a positive harmonic function is a contractible function in the hyperbolic metric. Moreover, if the equality sign holds in (6) for some pair of distinct points z and w, then the function u must be of the following form u(z) = Im(a Möbius transform of H onto H).Remark 1.2. The group of all conformal mappings of H onto itself is given byThus, a positive harmonic function u(z) is extremal for the inequality (6) if and only if it has the formwhere k > 0 and t ∈ R; here P (z, t) = 1 π y (x − t) 2 + y 2 , z = x + iy ∈ H, t ∈ R is the Poisson kernel for the upper half-plane. ON HARMONIC FUNCTIONS AND THE HYPERBOLIC METRIC 3 Proof of the resultLet Ω be any domain in C (or in R). A metric density ρ is any continuous function in Ω with nonnegative val...

Abstract. We prove representation theorems, the versions of Smirnov's theorem and Carathéo-dory type theorem for harmonic homeomorphisms of the unit disk onto Jordan surfaces with rectifiable boundaries. Further we establish the classical isoperimetric inequality and the Riesz-Zygmund inequality for Jordan harmonic surfaces without any smoothness assumptions on the boundary.

This paper deals with an extremal problem for harmonic functions in the unit ball of R n . We are concerned with the pointwise sharp estimates for the gradient of real-valued bounded harmonic functions. Our main result may be formulated as follows. The sharp constants in the estimates for the absolute value of the radial derivative and the modulus of the gradient of a bounded harmonic function coincide near the boundary of the unit ball. This result partially confirms a conjecture posed by D. Khavinson.2010 Mathematics Subject Classification. Primary 35B30; Secondary 35J05. Key words and phrases. bounded harmonic functions, estimates of the gradient, the Khavinson problem, the Schwarz lemma.

In this paper we give some results concerning Fréchet differentiable mappings between domains in normed spaces with controlled growth. The results are mainly motivated by Pavlović's equality for the Bloch semi-norm of continuously differentiable mappings in the Bloch class on the unit ball of the Euclidean space as well as the very recent Jocić's generalization of this result.