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Register a new userCharacterizations of Projective Hulls by Analytic Discs
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Barbara Drinovec Drnov\\v{s}ek
Publication date
2012
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English
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The notion of the projective hull of a compact set in a complex projective space was introduced by Harvey and Lawson in 2006. In this paper we describe the projective hull by Poletsky sequences of analytic discs, in analogy to the known descriptions of the holomorphic and the plurisubharmonic hull.
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We give conditions characterizing holomorphic and meromorphic functions in the unit disk of the complex plane in terms of certain weak forms of the maximum principle. Our work is directly inspired by recent results of John Wermer, and by the theory of the projective hull of a compact subset of complex projective space developed by Reese Harvey and Blaine Lawson.
In this article, we impose a new class of fractional analytic functions in the open unit disk. By considering this class, we define a fractional operator, which is generalized Salagean and Ruscheweyh differential operators. Moreover, by means of this operator, we introduce an interesting subclass of functions which are analytic and univalent. Furthermore, this effort covers coefficient bounds, distortions theorem, radii of starlikeness, convexity, bounded turning, extreme points and integral means inequalities of functions belongs to this class. Finally, applications involving certain fractional operators are illustrated.
For $frac12<p<infty$, $0<q<infty$ and a certain two-sided doubling weight $omega$, we characterize those inner functions $Theta$ for which $$|Theta|_{A^{p,q}_omega}^q=int_0^1 left(int_0^{2pi} |Theta(re^{itheta})|^p dthetaright)^{q/p} omega(r),dr<infty.$$ Then we show a modified version of this result for $pge q$. Moreover, two additional characterizations for inner functions whose derivative belongs to the Bergman space $A_omega^{p,p}$ are given.
Via a unified geometric approach, a class of generalized trigonometric functions with two parameters are analytically extended to maximal domains on which they are univalent. Some consequences are deduced concerning commutation with rotation, continuation beyond the domain of univalence, and periodicity.
Let $phi$ be a normalized convex function defined on open unit disk $mathbb{D}$. For a unified class of normalized analytic functions which satisfy the second order differential subordination $f(z)+ alpha z f(z) prec phi(z)$ for all $zin mathbb{D}$, we investigate the distortion theorem and growth theorem. Further, the bounds on initial logarithmic coefficients, inverse coefficient and the second Hankel determinant involving the inverse coefficients are examined.
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