An Application of Wilf's Subordinating Factor Sequence on Certain Subclasses of Analytic Functions
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1 International Journal of Applied Engineering Research ISS Volume 3 umber 6 (208 pp An Application of Wilf's Subordinating Factor Sequence on Certain Subclasses of Analytic Functions A. H. El-Qadeem D. A. Mohan 2 Department of Mathematics Faculty of Science Zagaig University Zagaig 4459 Al-Sharia Egypt. Abstract In this wor we derive several subordination results for a certain subclasses M ( ; A B ( ; A B of analytic functions defined on the open unit disc U. Maing use of the familiar principle of differential subordination several properties relationships involving the functions. Keywords phrases: analytic function Subordination principle Hadamard product subordinating factor sequence. 200 Mathematics Subject Classification: 30C45.. ITRODUCTIO Let A denoted to class of function of the form f ( a ( a 0 (. 2 which are analytic function in the open disc U { C: } Also let K denote the familiar class of functions f A which are also univalent convex in U. For 0 C ac are such that c a 0 Raina Sharma [5] defined the integral operator A A as following: J : (i for c a 0 a by Let M ( ; A B be the subclass of functions f A for which: ( J f ( A J f ( B ( ( B A 0 that is that (.5 ( J f ( J f ( M ( ; A B f A : U. ( J f ( (.6 B B ( A B ( ac J f ( Also for B A 0 let ( ; A B be the subclass of functions f A for which: ( J f ( A ( B ( J f ( form (.5 (.6 it is clear that a c a c ( ( ; ( ( ;. f A B f M A B (.7 It is easily to see that: (i M A B S A B ( ; ( ( ; ( see [ with p=]; A B K A B a c ( c ca a ( a ( c a 0 J f ( t t f ( t dt ; (.2 (ii M A B S A B (0; ( A B K A B (0; ( see [3]; (ii for a c by J f f ( ( (.3 where sts for Euler's Gamma function (which is valid for all complex numbers except the non-positive integers. For f ( defined by (. it is easily from (.2 (.3 that: ( c ( a ( ( a ( c a c J f a 2 c a ( 0. (.4 (iii M ( ; S ( ( ; C( the subclasses of starlie convex of order 0 type 0 see [2]; (iv M ( ; S ( ( ; K( the subclasses of starlie convex of order 0 see [6]. 2494
2 International Journal of Applied Engineering Research ISS Volume 3 umber 6 (208 pp Definition (Hadamard Product or Convolution. Given two functions f g A where f ( is given by (. g( is defined by such that f ( g ( w ( ( U g ( b 2 the Hadamard product (or convolution f (as usual by ( f g ( a b ( g f ( 2 g is defined Definition 2 (Subordination Principle. For two functions f g analytic in U we say that the function f ( is subordinate to g( in U write f g or f ( g ( ( U if there exists a Schwar function w( analytic in U with w (0 0 w ( ( U In particular if the function g is univalent in U the above subordination is equivalent to f (0 g (0 f ( U g ( U. Definition 3 (Subordinating Factor Sequence. A sequence b of complex numbers is said to be a subordinating factor sequence if whenever f ( of the form (. is analytic univalent convex in U we have the subordination given by a b ( f ( ( U ; a. MAI RESULT We will mae use of the following lemmas. Unless otherwise mentioned we assume in the reminder of this paper that 0 B A 0 a c R c a U. Lemma [4]. Let the function f( be given by (.. Then if ( ; f M A B ( c ( a ( B ( ( A B ( a ( A B (. ( a ( c 2 (2. Lemma 2 [4]. Let the function f ( be given by (.. Then if ( ; f A B ( c ( a ( B ( ( A B ( a ( A B (. ( a ( c 2 (2.2 Lemma 3 [7]. The sequence b is a subordinating factor sequence if only if 2 b 0 ( U Theorem. Let the function f ( defined by (. be in the class M ( ; A B. Then ( c ( a2 ( a ( B ( A B ( ( c2 ( f g ( g ( ( c ( a2 2 ( a ( B ( A B ( ( c2 ( A B ( (
3 International Journal of Applied Engineering Research ISS Volume 3 umber 6 (208 pp for every function g in K ( B ( A B ( ( A B ( f ( ( c ( a2 ( a ( c2 ( c ( a2 ( a ( B ( A B ( ( c2 ( U. (2.4 The following constant factor larger one. ( c ( a2 ( B ( A B ( ( a ( c2 ( c ( a 2 2 ( ( B ( A B ( ( A B ( a ( c2 in the subordination result (2.3 cannot be replaced by a Proof. Let let ( ; f M A B 2 Then we have g ( q K. ( c ( a2 ( B ( A B ( ( a ( c2 ( c ( a 2 2 ( ( B ( A B ( ( A B ( a ( c2 ( c ( a2 ( ( B ( A B ( a ( c2 ( c ( a2 2 ( ( B ( A B ( ( A B ( a ( c2 2 ( f g ( a q. (2.5 Thus by Definition 3 subordination result (2.3 will hold true if the sequence ( c ( a2 ( ( B ( A B ( a ( c2 ( c ( a 2 a 2 ( ( B ( A B ( ( A B ( a ( c2 is a subordinating factor sequence ( with of course a. In view of Lemma 3 this is equivalent to the following inequality ow since ( c ( a2 ( ( B ( A B ( a ( c2 ( c ( a2 ( ( B ( A B ( ( A B ( a ( c2 a 0 ( U. (3.7 ( c ( ( ( ( ( ( a B A B ( a ( c is an increasing function of ( 2 we have ( c ( a2 ( ( B ( A B ( a ( c2 ( c ( a2 a ( ( B ( A B ( ( A B ( a ( c2 ( c ( a2 ( B ( A B ( ( a ( c2 ( c ( a2 ( ( B ( A B ( ( A B ( a ( c2 ( c ( a 2 a ( ( ( ( ( ( a ( ( ( B A B A B B A B ( c 2 a c 2 ( c ( 2 ( a ( 2 ( c ( a2 ( B ( A B ( ( a ( c2 ( c ( a2 ( ( B ( A B ( a ( AB( ( c2 ( c ( a ( c ( a2 ( ( ( ( ( ( ( a ( B ( ( A B ( B A B A B ( c a r a ( c2 2 r ( c ( a2 ( B ( A B ( ( a ( c2 ( AB( ( c ( a ( ( B ( A B ( a 2 r ( c ( ( ( ( ( ( a 2 A B B A B ( A B ( ( c 2 ( a ( c 2 r 0 ( r r (
4 International Journal of Applied Engineering Research ISS Volume 3 umber 6 (208 pp where we have also made use of assertion (2. of Lemma. Thus (2.7 holds true in U. This proves the inequality (2.3. The inequality (2.4 follows from (2.3 by taing the convex function g (. 2 sharpness of the constant To prove the ( c ( a2 ( B ( A B ( ( a ( c2 ( c ( a 2 2 ( ( B ( A B ( ( A B ( a ( c2 we consider the function given by f M A B 0 ( ; ( A B( f 2 0 (. ( c ( a 2 ( a ( B ( A B ( ( c2 (2.8 Thus from (2.3 we have ( c ( a 2 ( B ( A B ( ( a ( c2 a f B A B A B 0 U ( ( 2 2 c ( ( ( ( ( ( a ( c2 (. (2.9 Moreover it can easily be verified for the function f ( 0 given by (2.8 that ( c ( a2 ( ( B ( A B ( a ( c2 min ( c ( a 2 f 2 ( ( ( ( ( ( 0( r B A B A B a ( c2 2 (2.0 see Figure. This shows that the constant ( c ( a2 ( B ( A B ( ( a ( c2 ( c ( a 2 2 ( ( B ( A B ( ( A B ( a ( c2 is the best possible. c in Theorem we have: Corollary. Let the function f ( defined by (. be in the subclass S ( A B Then suppose that g ( K. ( B ( A B ( ( f g ( g ( (2. 2 ( B 2( A B ( ( B 2( A B ( f ( ( U. ( B ( A B ( (2.2 Figure ( B ( A B ( 2 ( B 2( A B ( in the subordination result (2. cannot be replaced by a larger one. c 0 in Theorem we have: 2497
5 International Journal of Applied Engineering Research ISS Volume 3 umber 6 (208 pp Corollary 2. Let the function f ( defined by (. be in the subclass S ( A B Then suppose that g ( K. A 2B ( f g ( g ( 2 ( B 2( A B ( B 2( A B f ( ( U. A 2B A2B (2.3 (2.4 2 ( B 2( A B in the subordination result (2.3 cannot be replaced by a larger one. c A B in Theorem we have: Corollary 3. Let the function f ( defined by (. be in the subclass S ( ; Then suppose that g ( K. (32 ( f g ( g ( 2 (5 4 (2.5 (5 4 f ( ( U. (32 (32 (2.6 2 (54 in the subordination result (2.5 cannot be replaced by a larger one. c A B in Theorem we have: Corollary 4. Let the function f ( defined by (. be in the subclass S ( suppose that g ( K. Then 2 ( ( ( 2(3 2 f g g f 3 2 ( (. 2 U (2.7 (2.8 2 in the subordination result (2.7 2(3 2 cannot be replaced by a larger one. Theorem 2. Let the function f ( defined by (. be in the class ( c ( a2 ( a ( c2 ( c ( a2 ( a ( c2 ( ; A B. Then ( B ( A B ( ( f g ( g ( 2 ( B ( A B ( ( A B ( (2.9 for every function g in K 2 ( B ( A B ( ( A B ( f ( ( U. ( c ( a2 ( a ( c2 ( c ( a2 ( a 2 ( B ( A B ( ( c2 (2.20 The following constant factor be replaced by a larger one. ( c ( a2 ( B ( A B ( ( a ( c2 ( c ( a2 ( a ( c2 2 ( B ( A B ( ( A B ( in the subordination result (2.9 cannot c in Theorem 2 we have: Corollary 5. Let the function f ( defined by (. be in the subclass K ( A B suppose that g ( K. Then 2498
6 International Journal of Applied Engineering Research ISS Volume 3 umber 6 (208 pp ( B ( A B ( ( f g ( g ( 2( B 3( A B ( (2.2 2( B 3( A B ( f ( ( U. 2 ( B ( A B ( (2.22 The following constant factor ( B ( A B ( 2( B 3( A B ( in the subordination result (2.2 cannot be replaced by a larger one. c 0 in Theorem 2 we have: Corollary 6. Let the function f ( defined by (. be in the subclass K ( A B suppose that g ( K. Then A 2B ( f g ( g ( 3A 5B (2.23 3A 5B f ( ( U. 2 A 2B (2.24 A2B 3A5B in the subordination result (2.23 cannot be replaced by a larger one. c A B in Theorem 2 we have: Corollary 7. Let the function f ( defined by (. be in the subclass C ( ; suppose that g ( K. Then (32 ( f g ( g ( 2 (4 3 (4 3 f ( ( U. (3 2 (32 in the subordination result (2.25 cannot be replaced by a larger one. 2 (4 3 c A B in Theorem 2 we have: Corollary 8. Let the function f ( defined by (. be in the subclass K ( suppose that g ( K. Then 2 ( f g ( g ( 53 f 5 3 ( (. 2(2 U in the subordination result (2.27 cannot be replaced by a larger one. (2.25 (2.26 (2.27 (
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