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We obtain improved versions of a classical theorem of Rogosinski concerning the partial sums of a bounded holomorphic function defined on the open unit disk D . Further, we establish refined versions of a generalized Bohr inequality for holomorphic functions mapping D inside a simply connected or convex domain Ω ⊊ C . In addition, we improve on the classical Bohr inequality for the family of holomorphic self mappings of D and for its subfamily consisting of functions that fix the origin.

A classical theorem of L. Carleson states that the injection map from the Hardy space H p into L p ( d μ ) is bounded if and only if the positive measure μ on the unit disc is a bounded Carleson measure. In this paper Forelli–Rudin estimates for arbitrary positive measures on the unit disc are proved, and then these estimates are applied to characterize bounded s-Carleson measures in terms of α-Bloch- and F ( p , q , s ) -functions.

In this paper we generalize a version of the classical Calderón–Zygmund theorem on principle value integrals in generalized Lebesgue spaces Lp(⋅) proved in [J. Reine Angew. Math. 563 (2003) 197–220], to kernels, which do not satisfy standard estimates on Rd+1. This result will be used in part II of this paper to prove the classical theorem on halfspace estimates of Agmon, Douglis, and Nirenberg [Comm. Pure Appl. Math. 12 (1959) 623–727] for generalized Lebesgue spaces Lp(⋅).

Bernstein's classical theorem states that for a polynomialPof degree at mostn, max|z|=1|P′(z)|≤nmax|z|=1|P(z)|. We give related results for polynomialsPsatisfying the conditionsP′(0)=P″(0)=···=P(m−1)(0)=0 andP(z)≠0 for |z

Consider the renewal equation in the form (∗) u(t) = g(t) + ∝ o t u(t − τ) ƒ(τ) dτ , where ƒ(t) is a probability density on [0, ∞) and limt → ∞ g(t) = g0. Asymptotic solutions of (∗) are given in the case when f(t) has no expectation, i.e., ∝ 0 ∞ tƒ(t)dt = ∞ . These results complement the classical theorem of Feller under the assumption that f(t) possesses finite expectation.