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On Linear Combinations of n Analytic Functions


Affiliations
1 Department of Mathematics, University of Mysore Manasagangotri, Mysore-570006, India
2 Department of Mathematics, Maharaja's College, University of Mysore, Mysore-570005, India
     

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Let p ≥ 1 be an integer and let 0 ≤ α < p. An analytic function P(z) belongs to the class P(A,B,p,α) if P(z)=(p—α) P1 (z) + α where P1 (z) belongs to the class P (A,B) (-1 ≤ B < A ≤ 1) the well known Janowski class of analytic iunciions on |z| < l. Let f(z) = zp + Σakzk be analytic on | z | < 1 such that zf'(z)/f(z) = P(z) for some P(z) in the class P(A,B,p,α). We then say that f(z) belongs to the class S(A,B,p,α). In this note we consider linear combinations of analytic functions of the form Σ γj fj(z) where fj(z) {j=1,2.....,n) are in the class S(A,B,p,α) and γj n are complex numbers with Σ γj= 1 .We determine the radius of l n starlikeness and the radius of convexity of F(z) = Σ γjfj(z).

Keywords

Janowski Class, Starlike Functions, Radius of Convexity.
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  • On Linear Combinations of n Analytic Functions

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Authors

S. Nanjunda Rao
Department of Mathematics, University of Mysore Manasagangotri, Mysore-570006, India
S. Latha
Department of Mathematics, Maharaja's College, University of Mysore, Mysore-570005, India

Abstract


Let p ≥ 1 be an integer and let 0 ≤ α < p. An analytic function P(z) belongs to the class P(A,B,p,α) if P(z)=(p—α) P1 (z) + α where P1 (z) belongs to the class P (A,B) (-1 ≤ B < A ≤ 1) the well known Janowski class of analytic iunciions on |z| < l. Let f(z) = zp + Σakzk be analytic on | z | < 1 such that zf'(z)/f(z) = P(z) for some P(z) in the class P(A,B,p,α). We then say that f(z) belongs to the class S(A,B,p,α). In this note we consider linear combinations of analytic functions of the form Σ γj fj(z) where fj(z) {j=1,2.....,n) are in the class S(A,B,p,α) and γj n are complex numbers with Σ γj= 1 .We determine the radius of l n starlikeness and the radius of convexity of F(z) = Σ γjfj(z).

Keywords


Janowski Class, Starlike Functions, Radius of Convexity.