Certain inclusion properties of subclass of starlike and convex functions of positive order involving Hohlov operator

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1 Iteratioal Joural of Pure ad Applied Mathematial Siees. ISSN Volume 0, Number (207), pp Researh Idia Publiatios Certai ilusio properties of sublass of starlike ad ovex futios of positive order ivolvig Hohlov operator K.Thilagavathi Departmet of Mathematis Shool of advaed Siees VIT Uiversity, Vellore-63204, Idia. Abstrat I this paper we ivestigate some ilusio properties of sublasses of ovex ad starlike futios of positive order ivolvig Hohlov operator. Keywords ad Phrases: Srivastava-Wright ovolutio operator, Starlike futios, Covex futios, Uiformly Starlike futios, Uiformly Covex futios, Hadamard produt, Hohlov operator.. INTRODUCTION Let be the lass of futios aalyti i the uit disk. { z : z ad z } Let be the lass of futios f of the form f ( z) z az,. 2 The Gaussia hypergeometri futio F( a, b;, z ) give by ( a) ( b) F( a, b; ; z) z, ( z ) (.2) ( ) () 0 is the solutio of the homogeous hypergeometri differetial equatio z( z) w ( z) [ ( a b ) z] w ( z) abw( z) 0

2 86 K.Thilagavathi ad has rih appliatios i various fields suh as oformal mappigs, quasi oformal theory, otiued fratios, ad so o. Here, a, b, are omplex umbers suh that 0,, 2, 3,, ( a) 0 for a 0, ad for eah positive iteger, ( a) a( a )( a 2) ( a ) is the Pohhammer symbol. I the ase of k, k 0,,2,, F( a, b; ; z ) is defied if a j or b j k. We refer to[, 8] ad referees therei for some importat results. j where For f, we reall the operator Iab,,( f ) of Hohlov [4] whih maps ito itself defied by meas of Hadamard produt as I ab,,(f)(z) = zf(a, b; ; z) * f(z). (.3) It is a speial ase of Srivastava-Wright ovolutio operator. Therefore, for a futio f defied by (.) we have a, b, ( a) ( b) (.4) ( ) 2 () I ( f )( z) z a z. Usig the itegral represetatio, ( ) b b dt ( ) ( ) 0 a b b ( tz) F( a, b; ; z) t ( t), ( ) ( b) 0, We a write ( ) b b f ( tz) z [ Iab,,( f )]( z) t ( t) d t *. ( ) ( ) 0 a b b t ( tz) z Whe f() z equals the ovex futio z, the the operator Iab,,( f ) i this ase beomes zf( a, b; ; z ). If a, b, 2 with ( ) the the ovolutio operator I,,( f ) turs ito Berardi operator ab Bf ( z) [ Ia, b, ( f )]( z) t f ( t)d t. z 0 Ideed, I ( f ) ad I ( ),,2,2,3 f are kow as Alexader ad Libera operators, respetively.

3 Certai ilusio properties of sublass of starlike ad ovex futios 87 Let us deote (see [5], [6]) 2 8(aros k) for 0 k 2 2 ( k ) 8 P ( k) for k, 2 2 for k t ( t)( k ) ( t) (.5) where t (0,) is determied by k osh( ( t) /[4 ( t)]), is the Legedre's omplete Ellipti itegral of the first kid () t 0 dx ( x )( t x ) ad 2 ( t) ( t ) is the omplemetary itegral of () t. Let k be a domai suh that ad k k w u iv : u k ( u ) k v, 0 k. The domai k is ellipti for k, hyperboli whe 0k, paraboli whe k, ad a right half-plae whe k 0. If p is a aalyti futio with p(0) whih maps the uit dis oformally oto the regio k, the P( k) p(0). Pk ( ) is stritly dereasig futio of the variable k ad it values are iluded i the iterval (0,2]. Let f be of the form (.). If f k, the the followig oeffiiet iequalities hold true (f.[5]): ( P( k)) a, {}. (.6)! Similarly, if f of the form (.) belogs to the lass k, the (f., [6]) ( P( k)) a, {}. ( )! (.7) A futio f is said to be i the lass ( AB, ), ( \{0}, B A ), if it satisfies the iequality f ( z) ( A B) B[ f ( z) ], ( z ). (.8)

4 88 K.Thilagavathi The lass ( AB, ) was itrodued earlier by Dixit ad Pal [3]. Two of the may iterestig sublasses of the lass ( AB, ) by settig are worthy of metio here. First of all, i e os ( / 2 / 2), A 2 (0 ) ad B, the lass ( AB, ) Pousamy ad Roig [8], where redues essetially to the lass ( ) itrodued ad studied by ( ) : ( i ( R f e f ( z ) )) 0 ( z ; / 2 / 2,0 ). Seodly, if we put, A ad B (0 ), we obtai the lass of futios f satisfyig the iequality f ( z) ( z ;0 ) f ( z) whih was studied by (amog others) Padmaabha [7] ad Capliger ad Causey [2]. Motivated by the earlier work of Srivastava et al.[9], we itrodue two ew sublasses of amely M (, ) ad N (, ). we eed the followig results, to prove our mai results. Defiitio.. For some ( 0 ),we let M(, ) ad N (, ) be two ew sublass of osistig of futios of the form (.) with positive order 4 ( ) ad satisfyig the aalyti riteria 3 z( Iab,, f ( z)) M (, ) : f S :, z (.9) ( )( Ia, b, f ( z)) z( Ia, b, f ( z)) ad ( Ia, b, f ( z)) z( Ia, b, f ( z)) N(, ) : f S :, z ( Ia, b, f ( z)) z( Ia, b, f ( z)) respetively. (.0)

5 Certai ilusio properties of sublass of starlike ad ovex futios 89 COEFFICIENT BOUNDS: Lemma.2: A futio f of the form (.) belogs to the lass M(, ) if Ia, b, ( f ) / ( zia, b, ( f ) ad if ( a) ( b) ( ) a. ( ) 2 ( )! (.) Proof: Let f M (, ), the by (.9) z( I f ( z)) ab,, ' ' ( )( I f ( z)) z( I f ( z)) a, b, a, b, z( I f ( z)) ab,, ( )( I f ( z)) z( I f ( z)) a, b, a, b, ' ' (2 ), that is ( a) ( b) ( a) ( b) z a z ( ) a ( ) 2 () ( ) 2 () ( a) ( b) ( a) ( b) (2 ) (2 )( ) ( ) 2 () ( ) 2 () z a z a ( a) ( b) ( ) a () 2 () ( a) ( b) ( 2 ) z (2 )( ) a ( ) () 2 ( a) ( b) ( ) a ( ) 2 () ( a) ( b) [ a] 2 2( ) (2 )( ) ( ) () this implies ( a) ( b) ( ) a. (.2) ( ) 2 () Hee the theorem is proved.,

6 90 K.Thilagavathi Corollary. : Let f M (, ), the a 2 ( a) ( b) ( ) ( ) () Corollary.2: A futio f of the form (.) belogs to the lass N(, ) if I f zi a b f ad if ( ) / ( ' _,, ( )) ab,, ( a) ( b) ( ( ) ) a. (.3) ( ) 2 () Proof: It is well kow that f N (, ) if ad oly if zf ' M(, ). ( a) ( b) z( I f ( z)) z az, ( ) () Sie ' a, b, we may replae a with 2 a i Lemma (.2). Theorem.3. (CONVEX LINEAR COMBINATION) : The lass M (, ) is losed uder ovex liear ombiatio. Theorem.4 :The lass N (, ) is losed uder ovex liear ombiatio. Theorem.5( ARITHMETIC MEAN) Let f, f2,..., f defied by j, i, i 2 f ( z) a z, ( a 0, i,2,... l, 2) (.4) be i the lass M (, ). The the arithmeti mea of f ( z)( j,2,... l) defied by is also i the lass M (, ). l h( z) f j ( z), l (.5) j Theorem.6 : Let f, f2,..., f defied by

7 Certai ilusio properties of sublass of starlike ad ovex futios 9 j, i, i 2 f ( z) a z, ( a 0, i,2,... l, 2) (.6) be i the lass N (, ). The the arithmeti mea of f ( z)( j,2,... l) defied by is also i the lass N (, ).. l j h( z) f j ( z), l (.7) I the followig setio we obtai Ilusio results for the lasses M(, ), N(, ) usig Hohlov operator. 2. INCLUSION PROPERTIES Makig use of the followig lemma, we will study the atio of the hypergeometri futio o the lasses k, k. Lemma 2... (Dixit et al.[3] ) : If f ( A, B) is of form (.8), the a ( A B), {}. (2.) The result is sharp. Theorem 2.2: Let ab, {0}, a, b. Also, let be a real umber suh that a b. If f ( A, B), I ( f ) / ( zi ( f ) ad if the iequality ' a, b, a, b, ( ) ( a b ) ( ) ( )( a b ) ( a ) ( b ) ( a )( b ) ( ) ( ), ( A B) ( a )( b ) (2.2) is satisfied, the I,,( f) M(, ). Proof: ab Let f be of the form (.) belog to the lass ( AB, ). By virtue of Lemma (.2), it suffies to show that

8 92 K.Thilagavathi ( a) ( b) ( ) a. (2.3) ( ) 2 () Takig ito aout the iequality (2.) ad the relatio ( a) ( a ), we dedue that ( a) ( b) [ ( ) ] 2 ( ) () a ( a) ( A B) ( ) ( b) ( ) () 2 ( a ) ( b ) ( AB) ( ) ( ) 2 () ( a ) ( b ) ( A B) ( ) ( ) () 2 ( ) ( a ) ( b ) ( AB) ( ) ( a )( b ) ( ) () 2 ( ) ( )( A B) F( a, b, ;) ( ) ( AB) ( ) ( a )( b ) ( a )( b ) F( a, b, ;), here we use the relatio (a) = a(a+) -. (2.4) The required result ow follows by a appliatio of Gauss summatio theorem ad (2.2) ( a) ( b) ( ) a. (2.5) ( ) 2 () Theorem 2.3: Let Also, ab, {0}. Also, let be a real umber ad P ( ) P k be give by (.5). If f k UCV for some k (0 k < ) ad the iequality F F 3 2 a, b, P;, ; 3 2 a, b, P;, 2; 2, is satisfied, the Iab,,( f ) M(, ).. (2.6) Proof. Let f be give by (.). By (.), to show it is suffiiet I,,( f) M (, ). to prove that ab

9 Certai ilusio properties of sublass of starlike ad ovex futios 93 a b a (2.7) 2 Applyig the estimates for the oeffiiets give by (.7) ad makig use of the relatios (2.4) ad a a, we get a b a 2 a b P 2 a b P a b P 2 2 F F a b P a b P 3 2,, ;, ; 3 2,, ;, 2;, provided the oditio (2.6) is satisfied. Theorem 2.4. Let a, b C \ {0}. Also, let be a real umber suh that > a + b +. If f R A, B ad if the iequality a b a b ab a b, A B (2.8) is satisfied, the I,, ( f) N (, ). ab Proof. Let f be of the form (.) belog to the lass R A, B. By virtue of Corollary(.2), it suffies to show that a b a. (2.9) 2 Takig ito aout the iequality (2.) ad the relatio a a, we dedue that a b a 2

10 94 K.Thilagavathi a b a b A B A B 2 2 a b a b A B ab,, ;,, ; A B F a b F a b ab a b a b A B A B - a a a b a b a a A B A B ab a b A B A B A B α, provided the oditio (2.8) is satisfied. Theorem 2.5. Let a, b C \ {0}. Also, let be a real umber ad by (.5). If, for some k (0 k < ), f k UCV ad the iequality P P k be give ab P F 3 2 a, b, P;, 2; F a b P 3 2,, ;, ; 2, (2.0) is satisfied, the I,,( f) N (, ). ab Proof. Let f be give by (.). By (.) to show that I,,( f) N (, )., it is suffiiet to prove that a b a. 2 ab (2.) Applyig the estimates for the oeffiiets give by (.6) ad makig use of the relatios (2.4) ad a a, we get

11 Certai ilusio properties of sublass of starlike ad ovex futios 95 a b a 2 a b P 2 ab P a b P 2 a b P ab P F 3 2 a, b, P;, 2;, F a b P 3 2,, ;, ; provided the oditio (2.0) is satisfied Theorem 2.6: Let ab, {0}. Also, let be a real umber ad P ( ) P k be give by If f k, for some k(0 k ) ad the iequality ab P ab P ( ) 3F2( a, b, P ;,; ) (2 ) 3F2( a, b, P ;, 2;) ( ) ( ) F a, b, P ;,; 2( ), (2.2) 3 2 is satisfied, the I,,( f) N (, ). ab Proof: Let f be give by (.).We will repeat the method of provig used i the proof of Theorem (2.4). Applyig the estimates for the oeffiiets give by (.7),ad makig use of the relatios (2.4) ad ( a) ( a ), we get ( a) ( ) [ ( ) ] b a [ ( ) ( ) ] ( ) 2 () 2

12 96 K.Thilagavathi 2 ( a ) ( b ) ( P ) ( a ) ( b ) ( P ) ( ) ( )( ) ( ) ( ) () () ( ) () () 2 ( a ) ( b ) ( P ) [ ( )( ) ( ) ] ( ) () () ( a ) ( b ) ( P ) ( a ) ( b ) ( P ) ( ) ( ) ( ) () () ( ) () () ( a ) ( b ) ( P ) ( a ) ( b ) ( P ) ( ) ( ) ( ) () () ( ) () ( ) ( a ) ( b ) ( P ) ( a ) ( b ) ( P ) ( ) (2 ) ( ) () () ( ) () () ( ) ( a ) ( b ) ( P ) ( ) () () ( ) ab P ( ) 3F2 a, b, P ;,; ab P (2 ) 3F2 a, b, P ;, 2; ( ) 3F2( a, b, P ;,;), ( ) provided the oditio (2.2) is satisfied. ACKNOWLEDGEMENT: I express my siere thaks to my guide Prof. Dr. G. Murugusudaramoorthy for his valuable suggestios ad iformative remarks for presetig this paper. REFERENCES []. B. C. Carlso ad D. B. Shaffer, Starlike ad prestarlike hypergeometri futios, SIAM J. Math. Aal. 5(984) [2]. T. R. Capliger ad W. M. Causey, A lass of uivalet futios, Pro. Amer. Math. So. 39(973) [3]. K. K. Dixit ad S. K. Pal, O a lass of uivalet futios related to omplex order, Idia J. Pure Appl. Math. 26(9)(995) [4]. Y. E. Hohlov, Operators ad operatios i the lass of uivalet futios, Izv. Vyss Ueb. Zaved. Matematika, 0(978),83-89 (i Russia).

13 Certai ilusio properties of sublass of starlike ad ovex futios 97 [5]. S. Kaas ad A. Wisiowska, Coi regios ad k-uiform ovexity, J. Comput. Appl. Math. 05(999) [6]. S. Kaas ad A. Wisiowska, Coi regios ad k-starlike futios, Rev. Roumaie Math. Pures Appl. 45(2000) [7]. K. S. Padmaabha, O a ertai lass of futios whose derivatives have a positive real part i the uit dis, A. Polo. Math. 23(970) [8]. S. Pousamy ad F. Roig, Duality for Hadamard produts applied to ertai itegral trasforms, Complex Variables Theory Appl. 32(997) [9]. H. M. Srivastava, G. Murugusudaramoorthy ad S. Sivasubramaia, Hypergeometri futios i the paraboli starlike ad uiformly ovex domais, Itegral Trasform Spe. Fut. 8(2007)

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