Convolution And Coefficient Problems For Multivalent Functions Defined By Subordination - Supramaniam, Shamani (2009)

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Andaikan C satah kompleks, U = {z E C : Izl < I} cakera unit terbuka dalam C dan H(U) kelas fungsi analisis dalam U. Andaikan juga A kelas fungsi analisis 1 dalam U yang ternormalkan dengan 1(0) = 0 dan 1'(0) = 1. Fungsi 1 E A mempunyai siri Taylor berbentuk 00 l(z) = z + L anzn, (z E U). n=2 Andaikan Ap (p EN) kelas fungsi analisis 1 berbentuk 00 1(z) = zP + L anzn, (z E U) n=p+1 dengan A := AI. Pertimbangkan dua fungsi dalam Ap. Hasil darab Hadamard (atau konvolusi) untuk 1 dan 9 ialah fungsi 1 * 9 berbentuk 00 (J * g)(z) = zP + L anbnzn. n=p+1 Let C be the complex plane and U := {z E C : Izl < I} be the open unit disk in C and H(U) be the class of analytic functions defined in U. Also let A denote the class of all functions I analytic in the open unit disk U := {z E C : Izl < I}, and normalized by 1(0) = 0, and 1'(0) = 1. A function I E A has the Taylor series expansion of the form 00 I(z) = z + ~ (LnZn (z E U). n=2 Let Ap (p EN) be the class of all analytic functions of the form 00 fez) = zP + ~ (LnZn n=p+l with A:= AI. Consider two functions in Ap. The Hadamard product (or convolution) of I and 9 is the function I * 9 defined by 00 (J * g)(z) = zP + ~ anbnzn . "=p+l