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Displaying similar documents to “Injectivity of sections of convex harmonic mappings and convolution theorems”

Convolution conditions for bounded α -starlike functions of complex order

A. Y. Lashin (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let A be the class of analytic functions in the unit disc U of the complex plane with the normalization f ( 0 ) = f ' ( 0 ) - 1 = 0 . We introduce a subclass S M * ( α , b ) of A , which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class S M * ( n , α , b ) ( n 0 ) related to S M * ( α , b ) is also considered under the same conditions. Among other things, we find convolution conditions for a function f A to belong to the class S M * ( α , b ) . Several properties of the class S M * ( n , α , b ) are investigated. ...

On certain general integral operators of analytic functions

B. A. Frasin (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper, we obtain new sufficient conditions for the operators F α 1 , α 2 , . . . , α n , β ( z ) and G α 1 , α 2 , . . . , α n , β ( z ) to be univalent in the open unit disc 𝒰 , where the functions f 1 , f 2 , . . . , f n belong to the classes S * ( a , b ) and 𝒦 ( a , b ) . The order of convexity for the operators  F α 1 , α 2 , . . . , α n , β ( z ) and G α 1 , α 2 , . . . , α n , β ( z ) is also determined. Furthermore, and for β = 1 , we obtain sufficient conditions for the operators F n ( z ) and G n ( z ) to be in the class 𝒦 ( a , b ) . Several corollaries and consequences of the main results are also considered.

On a result by Clunie and Sheil-Small

Dariusz Partyka, Ken-ichi Sakan (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping F in the unit disk 𝔻 , if F ( 𝔻 ) is a convex domain, then the inequality | G ( z 2 ) - G ( z 1 ) | < | H ( z 2 ) - H ( z 1 ) | holds for all distinct points z 1 , z 2 𝔻 . Here H and G are holomorphic mappings in 𝔻 determined by F = H + G ¯ , up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain Ω in and improve it provided F is additionally a quasiconformal mapping...

More on exposed points and extremal points of convex sets in n and Hilbert space

Stoyu T. Barov (2023)

Commentationes Mathematicae Universitatis Carolinae

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Let 𝕍 be a separable real Hilbert space, k with k < dim 𝕍 , and let B be convex and closed in 𝕍 . Let 𝒫 be a collection of linear k -subspaces of 𝕍 . A point w B is called exposed by 𝒫 if there is a P 𝒫 so that ( w + P ) B = { w } . We show that, under some natural conditions, B can be reconstituted as the convex hull of the closure of all its exposed by 𝒫 points whenever 𝒫 is dense and G δ . In addition, we discuss the question when the set of exposed by some 𝒫 points forms a G δ -set.

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

A pure smoothness condition for Radó’s theorem for α -analytic functions

Abtin Daghighi, Frank Wikström (2016)

Czechoslovak Mathematical Journal

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Let Ω n be a bounded, simply connected -convex domain. Let α + n and let f be a function on Ω which is separately C 2 α j - 1 -smooth with respect to z j (by which we mean jointly C 2 α j - 1 -smooth with respect to Re z j , Im z j ). If f is α -analytic on Ω f - 1 ( 0 ) , then f is α -analytic on Ω . The result is well-known for the case α i = 1 , 1 i n , even when f a priori is only known to be continuous.

Sum-product theorems and incidence geometry

Mei-Chu Chang, Jozsef Solymosi (2007)

Journal of the European Mathematical Society

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In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P 1 , , P 4 , and Q 1 , , Q n 2 , if there are n ( 1 + δ ) / 2 many distinct lines between P i and Q j for all i , j , then P 1 , , P 4 are collinear. If the number of the distinct lines is < c n 1 / 2 then the cross ratio of the four points is algebraic. 2. Given c > 0 , there is δ > 0 such that for any P 1 , P 2 , P 3 2 noncollinear, and Q 1 , , Q n 2 , if there are c n 1 / 2 many distinct lines between P i and Q j for all i , j , then for any P 2 { P 1 , P 2 , P 3 } , we have δ n distinct lines between P and Q j . 3. Given...