On univalent functions, Bloch functions and VMOA

Pommerenke, Ch.

doi:10.1007/bf01351365

Cited by 101 publications

(84 citation statements)

References 21 publications

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“…Therefore (see for instances [18] and [23]), h is injective on ∆ and Ω = h(∆) is a convex domain. To get (ii) for n ≥ 3 we apply Ruscheweyh's generalization of the Schwarz-Pick lemma ( [20], see also [2]) to get the sharp estimates for the derivatives…”

Section: Proof the Condition (11) Immediatly Implies H(t) = H(z)mentioning

confidence: 99%

“…[14] and [18]), we have to consider in the analogon to Proposition 1 the functions g 1 subordinated to a function g ∈ C, the family of normalized close-to-convex functions and, naturally, replace 2 n−1 by (n + 1)2 n−2 . In the analogon to Proposition 2 we have to act likewise concerning g 1 and to replace the inequality (8) by…”

Section: Theorem 2 Let ω Be a Convex Proper Subdomain Of C And π Limentioning

confidence: 99%

“…This means that C \ Π is the union of closed halflines such that the corresponding open halflines are disjoint (compare f.i. [8], [18], and [14]). We prove…”

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confidence: 99%

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The punishing factors for convex pairs are $2^{n-1}$

Avkhadiev¹,

Wirths²

2007

Rev. Mat. Iberoam.

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Section: Proof the Condition (11) Immediatly Implies H(t) = H(z)mentioning

confidence: 99%

Section: Theorem 2 Let ω Be a Convex Proper Subdomain Of C And π Limentioning

confidence: 99%

See 1 more Smart Citation

The punishing factors for convex pairs are $2^{n-1}$

Avkhadiev¹,

Wirths²

2007

Rev. Mat. Iberoam.

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“…Since/(0) ^ oo, it follows from Lemma 6 that the point /(0) must lie on 3A(a, ß). Applying the Taylor expansion (9) f(z)=f(0) + akzk + ..., whereA:>l, we see that the local property of/at the origin is similar to that of zk. It follows that the angles ß -a and 2m -(ß -a) of A(a, ß) and A(a, ß)c, respectively, are mapped by / onto the angles k(ß -a) and k(2m -(ß -a)).…”

mentioning

confidence: 91%

On the generalized Seidel class 𝑈

Hwang¹

1983

Trans. Amer. Math. Soc.

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Abstract. As usual, we say that a function/ G U if /is meromorphic in | z | < 1 and has radial limits of modulus 1 a.e. (almost everywhere) on an arc A of | z \ = 1. This paper contains three main results: First, we extend our solution of A. J. Lohwater 's problem (1953) by showing that if / e U and/has a singular point P on A, and if v and \/v are a pair of values which are not in the range of /at P, then one of them is an asymptotic value of/at some point of A near P. Second, we extend our solution of J. L. Doob's problem (1935) from analytic functions to meromorphic functions, namely, if / £ U and /(0) = 0, then the range of / over | z | < 1 covers the interior of some circle of a precise radius depending only on the length of A. Finally, we introduce another class of functions. Each function in this class has radial limits lying on a finite number of rays a.e. on | z | = 1, and preserves a sector between domain and range. We study the boundary behaviour and the representation of functions in this class.

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“…For the exact mathematical formulation of the boundary-value problem which we intend to study, we need to characterize exactly the double-body boundary 8Db. We recall that a closed rectifiable Jordan curve dDB is said to be quasi-smooth [29], or of bounded arc-length-chord-length ratio [45], if t(w\,wi)l\w\ -W2I < b, 1 < b < 00, W],W2 G 8DB, (2.1) where £(wi,w2) is the length of the shorter arc of dDB between wx and w2. Also, a closed rectifiable Jordan curve 8Db is called of bounded rotation [31, p. 225], if the forward half-tangent exists at every point of dDB, and the tangent angle 1(5) which it makes with a fixed direction (the slope) may be defined as a function of bounded variation of the arc length s, 0 < s < L. to the class W{'p if: (i) it is quasi-smooth, and (ii) it is of bounded rotation, with jump index a < \/p.…”

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confidence: 99%

Untitled

1990

Quart. Appl. Math.

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Abstract. The two-dimensional, deep-water, wave-body interaction problem for a single-hulled body, floating on the free surface of an ideal liquid, is considered. The body boundary may be nonsmooth and may intersect the free surface at arbitrary angles. The existence of a unique solution representable by a multipole-series expansion is proved for all but a discrete set of oscillation frequencies. The proof is based on the property of the associated multipoles to be a basis of the space Lp(-n,0), 1 < p < 2. Strict estimates of the form Dn -0(n~a) are also obtained for the coefficients of the multipole-series expansion for piecewise smooth (0 < a < 2) and smooth (a = 2) body boundaries.1. Introduction. Consider an infinitely long, horizontal cylinder floating on the free surface of an unbounded, infinitely deep, incompressible and inviscid liquid. In this paper we are concerned with the solvability (well-posedness) of the boundary-value problems arising when the floating body is forced, by an incident harmonic wave or by an external force, to perform time-harmonic oscillations of small amplitude about its position of stable hydrostatic equilibrium.The uniqueness question for the wave-floating body interaction problem has been studied by means of two, essentially different, methodologies. The first one is of geometric character and appropriately uses Green's theorem to ensure that the only possible solution of the homogeneous problem is the trivial one. Such an approach was inaugurated by John [18] in 1950 (see also Lenoir [24], who reworked the twodimensional case, and Kleinman [21]). Recently, Simon and Ursell [33] essentially generalized John's uniqueness theorem in a manner permitting them to consider floating and/or submerged bodies of quite a general shape. In general, the uniqueness results obtained via this methodology are valid for all wavelengths but have suffered, up to now, from some geometrical restrictions concerning the shape of the floating body. Having established uniqueness, the well-posedness of the problem can be deduced by using various methods, such as the integral-equation formulation [18,21,22] or the limiting absorption principle [11,24,25].

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On univalent functions, Bloch functions and VMOA

Cited by 101 publications

References 21 publications

The punishing factors for convex pairs are $2^{n-1}$

The punishing factors for convex pairs are $2^{n-1}$

On the generalized Seidel class 𝑈

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