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Let $\Sigma_k(p)$ be the class of univalent meromorphic functions defined on the unit disc $\mathbb D$ with $k$-quasiconformal extension to the extended complex plane $\widehat{\mathbb C}$, where $0\leq k < 1$. Let $\Sigma_k^0(p)$ be the class of functions $f \in \Sigma_k(p)$ having expansion of the form $f(z)= 1/(z-p) + \sum_{n=1}^{\infty}b_n z^{n}$ on $\mathbb D.$ In this article, we obtain sharp area distortion and weighted area distortion inequalities for functions in $\Sigma_k^0(p)$. As a consequence of the obtained results, we present a sharp upper bound for the Hilbert transform of characteristic function of a Lebesgue measurable subset of $\mathbb D$.
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참고문헌 (18)
참고문헌 신청
K. Astala / 1994 / Area distortion of quasiconformal mappings / Acta Math 173(1):37~60
K. Astala / 2009 / Elliptic partial differential equations and quasiconformal mappings in the plane / Princeton University Press
K. Astala / 2003 / Composites and quasiconformal mappings: new optimal bounds in two dimensions / Calc. Var. Partial Differential Equations 18(4):335~355
F. G. Avkhadiev / 2007 / A proof of the Livingston conjecture / Forum Math 19(1):149~157
B. Bhowmik / 2009 / Concave functions, Blaschke products, and polygonal mappings / Sib. Math. J 50:609~615
K. Astala / 2009 / Elliptic partial differential equations and quasiconformal mappings in the plane / Princeton University Press
K. Astala / 2003 / Composites and quasiconformal mappings: new optimal bounds in two dimensions / Calc. Var. Partial Differential Equations 18(4):335~355
F. G. Avkhadiev / 2007 / A proof of the Livingston conjecture / Forum Math 19(1):149~157
B. Bhowmik / 2009 / Concave functions, Blaschke products, and polygonal mappings / Sib. Math. J 50:609~615
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