Asymptotically Sharp Bounds in the Hardy-Littlewood Inequalities on Mean Values of Analytic Functions

Let f be analytic in the unit disc, and let it belong to theHardy space H^p, equipped with the usual norm ||f||_p. It is knownfrom the work of Hardy and Littlewood that for q > _p, theconstants [formula] with the usual extension to the case where q = , have C(p,q)< . The authors prove that [formula] and [formula] 2000 Mathematics Subject Classification 30D55, 30A10.     

Sufficient conditions for one-sheetedness in nonconvex domains

Siberian Mathematical Journal -     

On Rellich’s Inequalities in Euclidean Spaces

On domains of Euclidean spaces we consider inequalities for test functions and their Laplacians. We describe a family of domains having vanishing Rellich constants. For the Euclidean space of dimension 4 we present a new version of the Rellich inequality. In addition, we prove new one-dimensional Rellich-type integral inequalities for linear combinations of test functions and their derivatives of orders one and two.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Versions of One Conformally Invariant Inequality

We obtain L_p-versions of theorems proved by J. L. Fernández and J. M. Rodríguez in the paper “The Exponent of Convergence of Riemann Surfaces, Bass Riemann Surfaces”, Ann. Acad. Sci. Fenn. Ser.A. I.Mathematica 15, 165–182 (1990). An important role in the proof of our results is due to the approach of V. M. Miklyukov and M. K. Vuorinen. In particular, we use the isoperimetric profile of hyperbolic domains.     

Schwarz—Pick Inequalities for Derivatives of Arbitrary Order

   Abstract. Let Ω and Π be two simply connected domains in the complex plane C which are not equal to the whole plane C and let λ _Ω and λ _Π denote the densities of the Poincare metric in Ω and Π , respectively. For f: Ω → Π analytic in Ω , inequalities of the type

Schwarz—Pick Inequalities for Derivatives of Arbitrary Order

Abstract. Let Ω and Π be two simply connected domains in the complex plane C which are not equal to the whole plane C and let λ _Ω and λ _Π denote the densities of the Poincare metric in Ω and Π , respectively. For f: Ω → Π analytic in Ω , inequalities of the type $$\frac{{|f^{(n)} (z)|}}{{n!}} \leqslant M_n (z,\Omega ,\Pi )\frac{{(\lambda _\Omega (z))^n }}{{\lambda _\Pi (f(z))}},z \in \Omega$$ are considered where M _n (z,Ω, Π) does not depend on f and represents the smallest value possible at this place. We prove that $$M_n (z,\Delta ,\Pi ) = (1 + |z|)^{n - 1}$$ if Δ is the unit disk and Π is a convex domain. This generalizes a result of St. Ruscheweyh. Furthermore, we show that $$C_n (\Omega ,\Pi ) = sup\left\{ {M_n (z,\Omega ,\Pi )|z \in \Omega } \right\} \leqslant 4^{n - 1}$$ holds for arbitrary simply connected domains whereas the inequality 2 ^n-1 ≤ C _n (Ω,Π) is proved only under some technical restrictions upon Ω and Π .     

Sufficient conditions for the univalence of quasiconformal mappings

Using two methods, quasiconformal continuation involving a theorem of Hadamard and direct estimation of f(z₂)?f(z₁), we obtain sufficient conditions for the univalence of continuously differentiable mappings f(z) of plane domains which, in the case of conformal mappings, reduce to both well-known and new results.     

Radii of convexity and close-to-convexity of certain integral representations

Strict upper bounds are determined for ¦s(z)¦, ¦Re s(z)¦, and ¦Im s(z) ¦ in the class of functions s(z)=a _nzⁿ+a_n+1zⁿ⁺¹+... (n¹) regular in ¦z¦<1 and satisfying the condition ¦u (₁) –u (₂) ¦K¦ ₁- ₂¦, where U()=Re s (eⁱ ), K>0, and ₁ and ₂ are arbitrary real numbers. These bounds are used in the determination of radii of convexity and close-to-convexity of certain integral representations.Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 581–592, May, 1970.The author wishes to thank L. A. Aksent'ev for his guidance in this work.     

Estimates of Hardy’s constants for tubular extensions of sets and domains with finite boundary moments

We study the behavior of constants in Hardy-type inequalities for tubular extensions of sets. We also obtain new estimates of constants for domains with finite boundary moments.