| Abstrakt: |
Seminal works of Hardy and Littlewood on the growth of analytic functions contain the comparison of the integral means M p (r , f) , M p (r , f ′) , M q (r , f) . For a complex-valued harmonic function f in the unit disk, using the notation | ∇ f | = (| f z | 2 + | f z ¯ | 2) 1 / 2 we explore the relation between M p (r , f) and M p (r , ∇ f) . We show that if | ∇ f | grows sufficiently slowly, then f is continuous on the closed unit disk and the boundary function satisfies a Lipschitz condition. We also prove that for 1 ≤ p < q ≤ ∞ , it is possible to give an estimate on the growth of M q (r , f) whenever the growth of M p (r , f) is known. We notably obtain Baernstein type inequalities for the major geometric subclasses of univalent harmonic mappings such as convex, starlike, close-to-convex, and convex in one direction functions. Some of these results are sharp. A growth estimate and a coefficient bound for the whole class of univalent harmonic mappings are given as well. [ABSTRACT FROM AUTHOR] |
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