Palestine Journal of Mathematics Vol. 2(1) (2013), Palestine Polytechnic University-PPU 2013

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1 Palestine Journal of Matheatics Vol. ( (03, Palestine Polytechnic University-PPU 03 On Subclasses of Multivalent Functions Defined by a Multiplier Operator Involving the Koatu Integral Operator Ajad S. Barha Ree A. Hadan Counicated by Ayan Badawi MSC 00 Classification: 37E45,37E30,37E40 Keywords phrases: Multivalent functions, Koatu integral operator. The author would lie to than the referee for his valuable suggestions coents. Abstract. This paper is devoted to the study of soe new subclasses strongly close-to-convex p-valent functions. It is defined by a ultiplier operator using the Koatu integral operator studies their inclusion relationships with the integral preserving properties.. INTRODUCTION A( pn, be the class of functions f ( of the for p f( = + a, ( pn, N (. = p+ n which are analytic p-valent in the open unit dis U = { : C, < }. The generalied Koatu integral operator K c, p: Apn (, Apn (, is defined for >0 c > p as ( c+ p c K c, pf ( = (log f ( t dt c t Γ( 0 t. (. Now, in ters of K c, p, we introduce the linear ultiplier operator J c, p,: Apn (, Apn (, as follows: 0,0 f ( = f ( (.3, f ( = ( K c, pf ( + ( K c, pf ( ' = f ( p,, f ( = ( f ( for >0, c > p, 0. If f Apn (, is given by (., then J p f ( = + B, (, a (.4 = p+ n where

2 87 Bahra Hadan c + p B, (, = ( p (.5 c + p for >0, c > p, 0. If f ( g( are analytic in U, we say that f ( is subordinate to g(, if a Schwar function w ( in U such that f( = gw ( ( exists then we write it as f g or f ( g(.. Results, (, AB, S η be the class of functions f Apn (, satisfying the condition ( f ( ' + A η p η f ( (.0 + B, then the following results appear: Lea.. h ( be convex univalent in U with h (0 = Re{ vh( + µ } > 0 ( v, µ. If p ( is analytic in U with p (0 =, then p '( p ( + h (, ( U which iplies p ( h (, ( U vp( + µ Lea.. h ( be convex univalent in U w ( be analytic in U with Re w ( 0. If p ( is analytic in U with p(0 = h(0, then p ( + w( p'( h (, ( U which iplies p ( h (, ( U. Lea.3. p ( be analytic in U with p (0 = p ( 0 in U. If two points exist, U such that for soeα, α ( α, α > 0 for all ( where α = arg p ( < arg p ( < arg p ( = α (. < =, then we have p '( ( α+ α = i p'( ( α+ α = i (. p ( p ( c α α c = i tan + c 4 α + α (.3 Proposition.. > 0, c > p, 0, h ( be convex univalent in U with h (0 = Re h ( > 0 ; if a function f( Apn (, satisfies the condition

3 88 On Subclasses of Multivalent Functions then f ( η h (, (0 η < ; U, p f ( f p f ( ( ( η, h ( (0 η < ; U. ( f ( J d( = p f ( where d( is analytic function in U ( c + p f ( cj f ( d( = p f ( ( c + p f ( = c p f ( p p ( c + p c + p + a + ( p p n ( c = + + p = c p p c + p + + ( p a = p+ n c + p then d (0 = (.4 Fro (.4, we get ( c + p J f ( ( p η d( + c + η = (.5 f ( Differentiating both sides logarithically with respect to ultiplying the by yields ( p d '( ( J f ( ( J f ( = ( p η d( + c + η f ( J f ( Dividing both sides by p η, we get (.6

4 89 Bahra Hadan d '( ( f ( + d( = ( p η d( + c + η p f ( By using Lea (., it follows that d ( h (,( U, then f p η f ( ( ( h (. (.7 Proposition.. h ( be a convex univalent in U with h (0 = Re( h ( > 0 ; if a function f( Apn (, satisfies the condition ( f ( J h (, (0 η < ; U, p f ( then ( Lf( J h (, (0 η < ; U, p Lf ( where L ( f is the integral operator defined by ( + L( f = Lf ( = t f ( t dt, ( 0 0 (.8 Now then p f( = + a ( pn, N = p+ n ( + p L f ( = t. t dt t at dt = p+ n + p + L f( = + a (.9 + p = p+ n + K c, p ( c + p c ( = log ( c Γ( t 0 L f t L f t dt ( c + p c + p + c = t log t dt a t log t dt c + Γ( t p 0 + = p+ n + t 0 K c, p + p + c + p L f( = + a (.0 + p = p+ n + c p = p+ n + + c + p ( c, p( L f ( K p c p = + + a

5 90 On Subclasses of Multivalent Functions By induction, ( L f ( = ( K c, p( L f ( + K, L f ( p ( c p(, + p + c + p ( Lf ( = + a [ + ( p] + p = p+ n + c + p J + p + c + p ( Lf ( = + + p a + p = p+ n + c + p, ( (. (, (, J,,, ( = c p ( + ( J ( ( L f f L f ( ( Lf( d( =, ( U (. p ( L f ( where d( is analytic function in U, with d (0 = Now ( ( Lf( J ( p η d( + η =, ( Lf ( f ( ( p η d( + η+ = ( + ( Lf ( Differentiating logarithically with respect to ultiplying by ( p d '( ( f ( ' + [( p η d + η] = + η+ ( p d( f ( Dividing both sides by ( p η adding subtracting η p d '( ( ( f ( ' + d( = η η ( p η d( p η, U + + f ( Therefore by Lea (., we obtain: d '( d ( + h ( c + η + ( p η d( ( ( Lf( ' d ( = [ ] h ( p ( Lf (, Corollary.. If f( S ( η, AB,, then operator defined by (.8., L( f S ( η, AB,, where L ( f is the integral

6 9 Bahra Hadan + A In Proposition (., tae h ( =, then + B ( f ( ' + A p η f ( + B ( ( Lf ( ' + A. p η ( L f ( + B Proposition.3. h ( be convex univalent in U with h (0 = Re( h ( > 0 ; if a function f( Apn (, satisfies the condition ( Lf( h ( η, (0 η < ; U p ( Lf ( ( ( Lf( J h (, (0 η < ; U (.3 p ( Lf ( Where L ( f is the integral operator defined by (.8.,, ( J,,( ( c p = ( + ( ( ( ( Lf c p Lf c Lf ( ( Lf( J d( = p ( Lf (, ( U (.4 Where d( is analytic function in U, with d (0 = Now ( ( Lf( J ( p η d( + η = ( Lf ( Differentiating logarithically with respect to ultiplying by ( p d '( ( ( Lf ( ' ( ( Lf ( ' = c+ η+ ( p d( J,, ( ( ( Lf( c p Lf J Dividing both sides by ( p η adding subtracting η p d '( ( ( Lf( ' + d( =,. c + η+ ( p d( p ( Lf ( d '( Therefore by Lea (., we obtain d ( + h ( c + η + ( p η d(

7 9 On Subclasses of Multivalent Functions ( ( Lf( ' d ( = h (. p ( Lf ( Theore.. f( Apn (, 0 <,, 0 γ <, if ( f ( ' < arg γ < J g(, for soe g ( S ( η, AB,, then ( f ( ' α < arg, γ < α g(, where α, α (0 < α, α, are the solutions of the equations : ( α+ α( c cos t α + tan (,for B ( p ( = η +Α ( + η+ c( + c + ( α+ α( c sin t +Β α,for B = (.5 ( α + α ( c cos t for B α tan ( + p η A = + ( + η+ c( + c + ( α+ α( c sin t α ( ( (.6 + B for B = where * C is given by (.3 ( p ( B t = sin ( p ( AB + ( η+ c( B (.7 Now ( f ( ' d( = γ p γ g(, ( U,,, = + ( J f ( ' ( c p J f ( cj f (,,,, (( p γ d( + γ J g( = ( J f ( ' = ( c + p J f ( cj f ( Differentiating both sides with respect to ultiplying by,, ( p γ d '( g( + [( p γ d( + γ]( g( ' = ( c + p( J f ( ' c( J f ( ' (.8

8 93 Bahra Hadan ( g( ' q ( = p g(, ( U Fro (.8 (.9, we have + ( J g( ' ( c p J g( ( p η q + η = = c,, g( g( ( c + p J g( ( p η q + η+ c = (.9 g(,, ( p γ d '( g( + [( p γ d( + γ]( g( ' Dividing both sides by ( p γ g( iplies = ( c + p( J f ( ' c( J f ( ' [( p γ d + γ] ( c + p ( J f ( ' [( p γ d( + γ] c d '( + (( p η q + η = ( p γ ( p γ g( ( p γ [( p γ d + γ] ( c + p ( J f ( ' d '( + (( p η q + η+ c = ( p γ ( p γ g( Dividing both sides by [( p η q + η+ c], then we get d '( ( J f ( ' + d( = γ ( p η q + η+ c ( p γ g( While, by using the result of Silveran Silvia [3], we have ( - AB ( A B q <, ( U; B - B B (.0 A Re{ q ( } >, ( U; B = (. fro (.0 (., we obtain ( p η q + η+ c = ρe Where ( p ( A ( p η( + A + η+ c < ρ < + η+ c ( B ( + B - t < φ < t for B where t is given by (.7, i ϕ ( p ( A + η+ c < ρ < < φ < for B =

9 94 On Subclasses of Multivalent Functions Here we note that d( is analytic in U with d (0 = Re( d( > 0 in U by applying the assuption Lea (. with w ( = ( p η q + η+ c. Here d( 0 in U. If the following two points exist:, U such that the condition (. is satisfied; then (by Lea.3 we obtain (. under the restriction (.3. At first, for the case B, we obtain: d'( arg( d( + ( p η q ( + η+ c + α = ( + B d'( arg( d( + ( p η q ( + η+ c * ( α α( c cos t tan ( p η( A * * + ( + η+ c( + c + ( α+ α( c sin t * ( α α( cos + c t α + tan ( p η( A * * = + ( + η+ c( + ( c + α+ α( c sin t ( + B where we have used inequality (.3,,, t are given by (.5, (.6 (.7 respectively. Siilarly, for the case B =, we obtain d '( arg( d( + α ( p η q ( + η+ c d '( arg( d( + α ( p η q ( + η+ c Which contradicts the assuption of the Theore, hence the proof is copleted. Corollary.. f( Apn (,, if where ( I p f ( ' arg γ < I p g( ( I pf ( ' arg γ < α I pg( I, p p ( J, p,0 ( = p+ n + p f = f = + a + (.

10 95 Bahra Hadan In Theore (., if we put =, c =, = 0, = = α = α = α, we get then I + p f = f = + a +, p p ( J, p,0 ( = p+ n ( I p f ( ' arg γ < I p g( ( I pf ( ' arg γ < α I pg( Theore.. f( Apn (, 0 <,, 0 γ <, if ( f ( ' < arg < γ, J g(, for soe g ( S ( η, AB,, then ( ( f ( J L α < arg γ < α, ( L g( where L ( f is defined by (.8, α, α (0 < α, α is the solutions of the equations ( α + α ( c cos t for B α tan ( + ( p η( A = + ( + η+ ( + c + ( α+ α( c sin t α + B ( α + α ( c cos t for B = for B α tan ( + ( p η( A = + ( + η+ ( + c + ( α+ α( c sin t α + B for B = (.3 (.4 Where c is given by (.3 t ( p ( A B = sin ( p ( AB + ( η+ ( B (.5

11 96 On Subclasses of Multivalent Functions ( ( f ( J L d( = γ, p γ ( Lg ( Fro Proposition (., Since g ( S, ( η, AB,, then by Corollary (., L( g S ( η, AB,, then [( p γ d( + γ] J ( Lg ( = ( ( Lf( ',, (, (, J, L ( = ( + ( J ( ( f f L f ( ( (( p γ d + γ J L g( = ( + J f ( J L f ( Differentiating both sides with respect to (( p γ d+ γ Lg ( + ( p γ d'( Lg ( = ( + J f( f(, ( (,, ( ( ( ( L Dividing both sides by ( J c, p, Lg ( ultiplying both sides by, then ( c p ( L ( ( c p ( L ( ( J Lg ( f ( f ( J J (( p γ d + γ + ( p γ d '( = ( + J Lg ( J Lg ( J Lg (,,,,,,,,, ( ( ( Lg ( J q ( = p ( Lg ( (( p γ d + γ( q( p η + η+ + ( p γ d '( = ( + ( ( L f ( ( Lg ( Now dividing both sides by ( p γ ( ( L f ( ( γ ( + ( d + ( q( p η + η+ + d '( = ( p γ ( p γ Lg ( Dividing both sides by ( q( p η + η+ d '( ( + ( J ( f ( d( + = γ ( q( p η + η+ ( p γ ( qp ( η + η+ J ( Lg (

12 97 Bahra Hadan fro (.0 (., we obtain where where t is given by (.5, d '( ( J f ( d( + = γ ( q( p η + η+ ( p γ g( ( p η q + η+ = ρe i ϕ ( p ( Α ( p η( +Α + η+ < ρ < + η+ ( Β ( +Β -t < φ < t for Β ( p ( Α + η+ < ρ < < φ < for Β= Here, we note that d( is analytic in U with d(0 = in U by applying the assuption Lea (. with ( = w ( p η q + η+ Hence, d( 0 in U if the following two points exist, U, such that the condition (. is satisfied then (by Lea.3, we obtain (. under the restriction (.3. At first, for the case B * ( α α ( cos t d '( + c arg( d( α + tan ( p q ( ( ( p η A * * η + η+ + ( + η+ ( + c + ( α+ α( c sin t ( + B = * ( α α( cos '( c t d + arg( ( d + α + tan ( p q( ( ( p η A * * η + η+ + ( + η+ ( + c + ( α+ α( c sin t ( + B = where we have used the inequality (.3,,, t are given by (.5, (.6 (.5 respectively. Siilarly, for the case B =, we obtain d'( arg( d( + α ( p η q ( + η+

13 98 On Subclasses of Multivalent Functions d'( arg( d( + α ( p η q ( + η+ These are contradiction to the assuption of Theore (.. This copletes the proof of the Theore (.. Corollary.3. f( Α( pn, 0 <, 0 γ < if, for soe g ( S ( η, AB,, then ( f ( ' arg( γ < g( ( L f ( ' arg( γ < α L g( where L ( f is defined by (.8, (0 < α is the solution of the equation cos α t tan α + = ( p η( + A + η+ + αsin t B + α for B for B = where t is given by (.5 Tae = = α = α = α in Theore (.,, for soe g ( S ( η, AB,, then ( f ( arg γ < g( ( ( f ( arg J L γ < α ( L g( where

14 99 Bahra Hadan cos α t tan α + = ( p η( + A + η+ + αsin t B + α for B for B = References [] A. Ebadian,S. Shas, Z.G. Wang Y. Sun, A class of ultivalent analytic functions involving the generalied Jung-Ki-Srivastava operator. Acta Univ. Apulensis 8, 65-77(009. [] F. M, Al-Oboudi, On univalent functions defined by a generalied Salagean operator. Int. J. Math. Math. Sci, 7, (004. [3] H. Silveran E. M. Silvia, Subclasses of starlie functions subordinate to convex functions, Canad.J. Math. 37, 48-6 (985. [4] M. E. Gordji, D. Aliohaadi A. Ebadian, Soe inequalities of the generalied Bernardi Libera-Livingston integral operator on univalent functions. J. Ineq. Pure Appl. Math. 0 (4, article 00, (009. [5] M. Nunoawa, S. Owa, H. Saitoh, N.E. Cho N. Taahashi, Soe properties of analytic functions at extreal points for arguents, preprint, 003. [6] M. S. Robertson, On the theory of univalent functions, Ann. Math. 37, (936. [7] P. Enigenberg, S. S. Miller, P.T. Mocanu M. O. Reade, On a Briot Bouquet differential subordination, in : General Inequalities, Vol. 3, Birhauser, Basel, (983. [8] S. G. Salagean, Subclasses of univalent functions, Lecture Notes in Math. 03, Springer- Verlag, Berlin, Heidelberg New Yor, 36-37, (983. [9] S. S. Miller P.T. Mocanu, Differential subordinations univalent functions, Michigan Math. J. 8, 57-7 (98. [0] Y. Koatu, On analytic prolongation of a faily of integral operators. Matheatica (Cluj 3, 4-45 (990. Author inforation Ajad S. Barha Ree A. Hadan, Departent of Matheatics, Palestine Polytechnic University, Hebron, Palestine. E-ail: ajad@ppu.edu, reeowaidat@yahoo.co Received: June 4, 0. Accepted: Septeber 7, 0.