Iteratioal Joural of Pure ad Applied Mathematial Siees. ISSN 0972-9828 Volume 0, Number (207), pp. 85-97 Researh Idia Publiatios http://www.ripubliatio.om 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
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.
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 2 2 4 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 2 2 2 ( x )( t x ) ad 2 ( t) ( t ) is the omplemetary itegral of () t. Let k be a domai suh that ad k 2 2 2 2 2 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)
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)
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.,
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
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
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
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
94 K.Thilagavathi a b a b A B A B 2 2 a b a b A B 2 2 2 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
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 2 2 2 2 2 2 2 2 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
96 K.Thilagavathi 2 ( a ) ( b ) ( P ) ( a ) ( b ) ( P ) ( ) ( )( ) ( ) ( ) () () ( ) () () 2 ( a ) ( b ) ( P ) [ ( )( ) ( ) ] ( ) () () ( a ) ( b ) ( P ) ( a ) ( b ) ( P ) ( ) ( ) ( ) () () ( ) () () 2 2 2 2 2 2 ( a ) ( b ) ( P ) ( a ) ( b ) ( P ) ( ) ( ) ( ) () () ( ) () ( ) 2 2 2 ( a ) ( b ) ( P ) ( a ) ( b ) ( P ) ( ) (2 ) ( ) () () ( ) () () ( ) 2 2 2 2 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) 737 745. [2]. T. R. Capliger ad W. M. Causey, A lass of uivalet futios, Pro. Amer. Math. So. 39(973) 357-36. [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) 889 896. [4]. Y. E. Hohlov, Operators ad operatios i the lass of uivalet futios, Izv. Vyss Ueb. Zaved. Matematika, 0(978),83-89 (i Russia).
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) 327 336. [6]. S. Kaas ad A. Wisiowska, Coi regios ad k-starlike futios, Rev. Roumaie Math. Pures Appl. 45(2000) 647 657. [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) 73-8. [8]. S. Pousamy ad F. Roig, Duality for Hadamard produts applied to ertai itegral trasforms, Complex Variables Theory Appl. 32(997) 263 287. [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) 5-520.
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