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DEMONSTRATIO MATHEMATICAVol. XXXVNo 22002F. M. Al-Oboudi, M. M. HidanDISPROOF OF A N INCLUSION RELATION OF CLASSESRELATED TO SPIRALLIKE FUNCTIONSAbstract. An inclusion relation between classes of functions, defined by Ruscheweyhderivative and related to spirallike functions, given by S.S. Bhoosnurmath andM. V. Devadas [2] is disproved and some new results axe given.1. IntroductionLet H denote the class of functions / which are analytic in the unit diskA — {z : \z\ < 1} and normalized by /(0) = 0 = /'(0) - 1. An analyticfunction / on A is said to be subordinate to an analytic function g on A(written / -< g) if f ( z ) = g(w(z)), z G A for some analytic function w withu>(0) = 0 and| w(z) |< 1 in A. The Hadamard product (convolution) of twopower seriesooooa zkf(z)=k >bkzk,9(z) =fc=0k=0is defined as the power seriesoo{f*9)(z) = ^2akbkz k ,z e A.fc=0Denote by Dn : H —• H the operator defined byD nf (z) = (1-Zz)n+I*f>neNo ={0,1,2,...}1991 Mathematics Subject Classification: Primary 30C45.Key words and phrases: analytic, Ruscheweyh derivative, convolution, spirallike, subordination.268F. M. A l - O b o u d i , M. M. HidanNote that D°f(z) = f(z) and D1f(z) = zf'{z). The symbol
Demonstratio Mathematica – de Gruyter
Published: Apr 1, 2002
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