ASIAN JOURNAL OF MATHEMATICS AND APPLICATIONS Volume 14, Article ID ama17, 13 ages ISSN 37-7743 htt://scienceasiaasia SECOND HANKEL DETERMINANT FOR SUBCLASSES OF PASCU CLASSES OF ANALYTIC FUNCTIONS M S SAROA, GURMEET SINGH AND GAGANDEEP SINGH Abstract: We define two subclasses of the class of Pascu functions For any real μ, we are interested in determining the uer bound of belonging to these classes aa 3 µ a for an analytic function f( z z az az 3 4 3 = ( z < 1 1 INTRODUCTION AND DEFINITION: PRINCIPLES OF SUBORDINATION: Let f ( z and F( z be two analytic functions in the unit disc E = { z: z < 1} Then, f ( z is said to be subordinate to F( z in the unit disc E if there exists an analytic function w( z in E satisfying the condition w ( =, w( z < 1 such that f ( z = F( w( z and we write as f ( z F( z In articular if F( z is univalent in D, the above definition is equivalent to f ( = F( and f ( E F( E FUNCTIONS WITH POSITIVE REAL PART: Let Ƥ denotes the class of analytic functions of the form (11 P( z z z = 1 1 with Re P( z >, z D Let A denote the class of functions of the form (1 ( f z = z az k k= k which are analytic in the oen unit disc D = { z: z < 1} S is the class of functions of the form (1 which are univalent The Hankel determinant: ([9],[1] Let f( z ( n = az be analytic in D For 1, n n= a a a n n 1 n q 1 H n = a a a q n 1 n n q a a a n q 1 n q n q q the qth Hankel determinant is defined by 1 Mathematics Subject Classification: 3C45 Key words and hrases: Subordination, Hankel determinant, functions with ositive real art, univalent functions and close to star functions 14 Science Asia 1 / 13
/ 13 M S SAROA, GS BHATIA AND GAGANDEEP SINGH The Hankel determinant was studied by various authors including Hayman[3] and Ch Pommerenke ([13],[14] For q= and n =, the second Hankel determinant for the analytic function f ( z is defined by a a 3 ( = = ( H aa a a a 4 3 3 4 R reresents the class of functions f ( z (13 ( f z Re >, z D z A and satisfying the condition R is a articular case of the class of close to star function defined by Reade[17] The class R and its subclasses were vastly studied by several authors including Mac-Gregor[7] Let R be the class of functions f ( z (14 Re f ( z, > z D Aand satisfying The class R was introduced by Noshiro [11] and Warschawski[18] ( known as N-W class and it was shown by them that R is a class of univalent functions The class R and its subclasses were investigated by various authors including Goel and the author ([1], [] For, R α denote the classes of functions in A which satisfy, resectively, α R1 ( α and ( the conditions f ( z (15 Re ( 1 α αf ( z >, z D z and (16 Re f ( z α zf ( z >, z D R α were introduced by Pascu [1] and are called Pascu classes of The classes R1 ( α and ( functions It is obvious that f ( z R1 ( α imlies that zf ( z ( R α We shall deal with the following classes f ( z 1 Az R1 ( α AB, = f A: ( 1 α αf ( z, α, 1 B< A 1, z D z 1 Bz (17 and 1 Az (18 R ( α A, B = f A : f ( z αzf ( z, α, 1 B < A 1, z D 1 Bz R1 ( α1, 1 R1 ( α and R ( α1, 1 R ( α R1 ( α AB, is a subclass of R1 ( α and R ( α AB, is a sublass of R ( α The classes R1 ( α AB, and R ( α AB, were studied by the author[8]througout the aer, we assume that α, 1 B < A 1 and z D PRELIMINARY LEMMAS
SECOND HANKEL DETERMINANT 3 / 13 Lemma 1 [15] Let P( z Ƥ ( z, then ( n = 1,,3, n Lemma [5] Let P( z Ƥ ( z, then ( = 4 x, 1 1 ( ( ( 3 4 = 4 x 4 x 4 1 x z 3 1 1 1 1 1 1 for some x and z with x 1, z 1 and, 1 3 MAIN RESULTS Theorem 31: Let f R1 ( α AB,, then aa µ a 4 3 (31 (3 (33 (34 { 3( 1 α µ ( 1 α( 1 } ( α( α( α ( α µ ( α( α 1 B 4µ A B 1 1 3 1 { 1 1 1 3 } ( 1 α { 3( 1 α 4µ ( 1 α( 1 1 B } 4µ A B ( 1 α( 1 ( 1 α {( 1 α µ ( 1 α( 1 } ( 1 α if µ 41 1 3 31 ( α ( α( α 1 B 4µ 31 ( α 31 ( α if µ A B ( 1 α 4( 1 α( 1 ( 1 α( 1 { µ ( 1 α( 1 3( 1 α 1 B } 4µ A B ( 1 α( 1 ( 1 α µ ( 1 α( 1 ( 1 α if µ 1 1 3 31 ( α ( α( α { } ( 1 α if µ Proof By definition of subordination, ( 1 α ( ( ( f z 1 Aw z αf ( z =, z 1 Bw z Taking real arts, ( ( ( f z 1 Aw z Re ( 1 α αf ( z = Re z 1 Bw z
4 / 13 M S SAROA, GS BHATIA AND GAGANDEEP SINGH which imlies that 1 B A B 1 Ar 1 A > 1 Br 1 B ( z = r 3 (35 1 ( 1 α az ( 1 α az 3 ( 1 3 α az = P 4 ( z Equating the coefficients in (35, we get (36 1 B a = 1 B a = 3 A B 1 B a = 1 A B ( 1 α ( 1 α 3 4 A B ( 1 System (36 ensures that (37 C( α ( aa µ a 4 3 ( 1 α (4 µ 1 3 ( 1 α( 1 ( =, A B (38 C ( α = 4 ( 1 α( 1 ( 1 α 1 B Using Lemma in (37, we obtain 3 ( α ( µ ( α 4 3 1 1 1( 1 1( 1 ( 1 C a a a = 1 4 x 4 x 4 1 x z ( 1 ( 1 3 1 ( 4 1 µ α α x for some x and z with x 1, z 1 or (39 ( α ( µ = ( 1 α µ ( 1 α( 1 C aa a 4 4 3 1 ( α µ ( α( α 1 ( 1 1 1 1 3 4 x ( 4 { ( 1 α µ ( 1 α( 1 } 4µ ( 1 α( 1 x 1 ( α 1( 4 1 1 x z 1 1 Relacing by [,] and alying triangular inequality to (39, we get 1 ( 1 ( 1 ( 1 3 4 ( 1 ( 1 ( 1 3 ( 4 α µ α α α µ α α δ C( α aa µ a 4 3 ( 4 ( 1 α µ ( 1 α( 1 4µ ( 1 α( 1 δ 1 ( α ( 4 ( 1 δ,( δ = x 1 which can be ut in the form
SECOND HANKEL DETERMINANT 5 / 13 ( (31 Cα aa µ a F ( 1 α µ ( 1 α( 1 4 ( 1 α ( 4 ( 1 α µ ( 1 α( 1 ( 4 δ ( 4 ( 1 α ( ( 4 3 ( 4 { ( 1 α µ ( 1 α( 1 } 4µ ( 1 α( 1 ( 1 α δ if µ ( 1 α µ ( 1 α( 1 4 ( 1 α ( 4 ( 1 α µ ( 1 α( 1 ( 4 δ { µ 1 α 1 } 4µ ( 1 α( 1 ( 1 α δ (1 α if µ ( 1 α( 1 µ ( 1 α( 1 ( 1 α 4 ( 1 α ( 4 µ ( 1 α( 1 ( 1 α ( 4 δ ( 4 { µ ( 1 α( 1 ( 1 α } 4µ ( 1 α( 1 ( 1 α δ if µ = F ( δ ( 1 α ( 1 α( 1 ( δ > and therefore F ( δ is increasing in [,1] ( (31 reduces to F δ attains its maximum value at 1 ( 1 α µ ( 1 α( 1 4 ( 1 α µ ( 1 α( 1 ( 4 δ = ( 4 {( 1 α µ ( 1 α( 1 } 4µ ( 1 α( 1 if µ ( 1 α µ ( 1 α( 1 4 ( 1 α µ ( 1 α( 1 ( 4 ( α µ { α µ α α } µ ( α( α µ (1 α ( 4 ( 1 ( 1 ( 1 3 4 1 1 3 ( 1 α( 1 C aa a 4 3 if µ ( 1 α( 1 ( 1 α 4 µ ( 1 α( 1 ( 1 α ( 4 ( 4 { µ ( 1 α( 1 ( 1 α } 4µ ( 1 α( 1 3 α, if µ ( 1 α ( 1 α( 1
6 / 13 M S SAROA, GS BHATIA AND GAGANDEEP SINGH = G(, or (311 C( α aa µ a 4 3 ( Case (i µ max G, ( = ( 1 ( 1 ( 1 3 4 4 3( 1 ( 1 ( 1 3 16( 1 ( 1 3 G α µ α α α µ α α α α µ G( is maximum for (31 Case (ii ( ( α µ ( α( α 3 ( α µ ( α( α G = 8 1 1 1 3 8 3 1 1 1 3 = which imlies that = ( α µ ( α( α 3 1 1 1 3 ( 1 α µ ( 1 α( 1 Putting the corresonding value of G( along with ( µ (1 α ( 1 α( 1 C α from (38 in (311, we get ( = ( 1 ( 1 ( 1 3 4 4 3( 1 4 ( 1 ( 1 3 16( 1 ( 1 3 G α µ α α α µ α α α α µ Sub-case (a 3(1 α µ 41 1 3 ( α( α It is easy to see that G( is maximum at = ( α µ ( α( α 3 1 4 1 1 3 ( 1 α µ ( 1 α( 1 Substituting the corresonding value of G( and the value of C ( α in (311, (3 follows Sub-case (b G 3(1 α ( α( α ( 1 α( 1 41 1 3 ( < and ( µ (1 α G is maximum at = In this sub-case max ( 16( 1 ( 1 3 G = α α µ Case (iii µ (1 α ( 1 α( 1 ( = ( 1 ( 1 3 ( 1 4 4 ( 1 ( 1 3 3( 1 16( 1 ( 1 3 G µ α α α µ α α α α α µ Sub-case (a (1 α µ 3(1 α ( 1 α( 1 1 ( α( 1
SECOND HANKEL DETERMINANT 7 / 13 G ( < and maximum ( = ( = 16( 1 ( 1 3 G G α α µ Combining the cases (ii-(b and (iii-(a we arrive at (33 Sub-case (b 3(1 α µ 1 1 3 ( α( α A simle calculus shows that G( is maximum at = Substituting the corresonding value of G( and the value of ( ( ( ( µ 1 α 1 3 1 α µ ( 1 α( 1 ( 1 α C α in (311, (34 follows Remark 31 Put A=1 and B= -1 in the theorem we get the estimates for the class R1 ( α Taking A = 1, B = -1 and α = in the theorem we have Corollary 31 If f R, then aa µ a 4 3 ( 3 µ 1 ( µ ( µ 1 ( µ 4 µµ, 3 4 3 4 µ, µ 4 3 3 4 µ, µ 4 ( µ 3 3 4 µµ, ( µ 1 Letting A = 1,B = -1 and α = 1 we get Corollary 3 If f R, then aa µ a 4 3 ( 144( 9 8µ ( 144( 9 8µ 7 16µ 4µ µ 9 7 3µ 4µ 7, µ 9 3 4µ 7 7, µ 9 3 16 ( 16µ 7 4µ 7, µ 144( 8µ 9 9 16 This results was roved by Janteng et al [4] Theorem 3 Let f R ( α AB, aa µ a 4 3, then
8 / 13 M S SAROA, GS BHATIA AND GAGANDEEP SINGH (31 (313 (314 (315 Proof We have ( ( 1 Aw z f ( z α zf ( z = 1 Bw z Taking real arts, ( ( 1 Aw z 1 1 Re Ar A f ( z α zf ( z = Re > 1 Bw z 1 Br 1 B This imlies that ( z = r 1 B A B 3 (316 1 ( 1 α az 3 ( 1 α az 4 3 ( 1 3 α az = P 4 ( z Identifying the terms in (316, we get (317 System (317 yields { 7( 1 α 16µ ( 1 α( 1 } µ ( α( α( α { ( α µ ( α( α } 91 ( α 1 B 4 A B 144 1 1 3 1 9 1 8 1 1 3 if µ { 7( 1 α 3µ ( 1 α( 1 1 B } 4µ A B 144( 1 α( 1 ( 1 α { 9( 1 α 8µ ( 1 α( 1 } 91 ( α if µ 3 1 1 3 7( 1 α ( α( α 1 B 4µ A B 91 ( α 7( 1 α 7( 1 α if µ 3 1 α 1 16 1 α 1 ( ( 7( 1 α ( α( α ( ( { 16µ ( 1 α( 1 7( 1 α 1 B } 4µ A B 144( 1 α( 1 ( 1 α 8µ ( 1 α( 1 9( 1 α A B 1 a = 1 B 1 A B a = 3 1 B 31 A B 3 a = 4 1 B 41 3 ( α ( α ( α { } 91 ( α if µ 16 1 1 3 (318 C( α ( aa µ a 4 3 =9 ( 1 α ( 4 8µ ( 1 α ( 1 ( 1 3,
SECOND HANKEL DETERMINANT 9 / 13 C B A 1 B (319 C ( α = 88( 1 α( 1 ( 1 α By Lemma, (319 can be written as 3 ( α ( aa4 µ a4 = 9( 1 α 1[ 1 1 ( 4 1 x 1 ( 4 1 x ( 4 1 ( 1 x z] ( ( 1 ( 1 8µ 1 α 1 4 x for some x and z with x 1, z 1 or (3 C( α ( aa µ a 4 4 3 9( 1 α 8µ ( 1 α( 1 = 1 9( 1 α 8µ ( 1 α( 1 ( 4 1 1 ( 4 9( 1 α 1 8µ ( 1 α( 1 3µ ( 1 α( 1 { } ( α 1( 1 18 1 4 1 x z Relacing by, 1 and alying triangular inequality to (3, we get ( α µ 4 3 ( α µ ( α( α C aa a 9 1 8 1 1 3 ( ( ( ( 9 1 α 8µ 1 α 1 4 δ 4 ( 4 9( 1 8 ( 1 ( 1 3 3 ( 1 ( 1 3 x x α µ α α µ α α δ ( α ( which can be ut in the form 18 1 4 1 δ ( δ = x 1
1 / 13 M S SAROA, GS BHATIA AND GAGANDEEP SINGH ( Cα aa µ a 4 3 9( 1 α 8µ ( 1 α( 1 4 18( 1 α ( 4 9( 1 α 8µ ( 1 α( 1 ( 4 δ ( ( ( ( { 9 1 8 1 1 3 } 3 ( 1 ( 1 3 18( 1 α α µ α α µ α α δ if µ 4 9( 1 α 8µ ( 1 α( 1 4 18( 1 α ( 4 9( 1 α 8µ ( 1 α( 1 ( 4 δ ( ( ( ( { 9 1 α 8µ 1 α 1 } 3µ ( 1 α( 1 4 δ 18( 1 α if µ 81 1 3 91 ( α ( α( α 8µ ( 1 α( 1 9( 1 α 4 18( 1 α ( 4 8µ ( 1 α( 1 9( 1 α ( 4 δ ( ( ( ( { 8µ 1 α 1 9 1 α } 3µ ( 1 α( 1 4 18( 1 α 91 ( α if µ 81 ( α( 1 = ( F δ δ (31 reduces to F ( δ > which means that F ( δ is increasing in [, 1] and F ( δ attains maximum value at δ = 1
SECOND HANKEL DETERMINANT 11 / 13 ( α ( µ 4 3 C aa a ( α µ ( α( α 4 ( α µ ( α( α ( 9 1 8 1 1 3 9 1 8 1 1 3 4 ( 4 { 9( 1 α 8µ ( 1 α( 1 } 3µ ( 1 α( 1 3 α, if µ 9 1 8 1 1 3 9 1 8 1 1 3 4 (4 9 1 8 1 1 3 3 1 1 if ( α µ ( α( α 4 ( α µ ( α( α ( { ( α µ ( α( α } µ ( α ( 91 ( α µ 81 ( α( 1 8µ 1 α 1 9 1 α 8µ 1 α 1 9 1 α 4 ( 4 { 8µ ( 1 α( 1 9( 1 α } 3µ ( 1 α( 1 91 ( α if µ 81 ( α( 1 ( ( ( 4 ( ( ( ( Or (3 C( α ( aa µ a 4 3 G( Case (i µ ( = 9( 1 α 8µ ( 1 α( 1 4 7( 1 α 16µ ( 1 α( 1 G 18 1 α 1 µ ( ( G( is maximum for G ( = 8 9( 1 α 8µ ( 1 α ( 1 which gives 4 7( 1 α 16µ ( 1 α( 1 = 9( 1 α 8µ ( 1 α( 1 Putting the corresonding value of G( along with the value of C ( α from (319 in (3, we get (31 3 [ ] 8[ 7( 1 α 16µ ( 1 α ( 1 ] = ( ( ( 91 α Case (ii µ 81 α 1 ( = 9( 1 α 8µ ( 1 α( 1 4 7( 1 α 3µ ( 1 α( 1 ( α( α µ G 18 1 1 3 4
1 / 13 M S SAROA, GS BHATIA AND GAGANDEEP SINGH Sub-case (a 7( 1 α ( α( α µ 3 1 1 3 An elementary calculus shows that G( is maximum at = ( α µ ( α( α ( α µ ( α( α 7 1 3 1 1 3 9 1 8 1 1 3 With the corresonding value of G( along with the value of C ( α in (3, we arrive at (313 Sub-case (b G ( α ( ( ( α ( ( 71 91 µ 3 1 α 1 8 1 α 1 ( < and ( G is maximum at = max ( ( 18( 1 ( 1 3 91 α Case (iii µ 81 α 1 G = G = α α µ ( ( ( ( = 8µ ( 1 α( 1 9( 1 α 4 16µ ( 1 α( 1 7( 1 α G 18 1 α 1 µ ( ( 4 Sub-case (a G ( α ( ( ( α ( ( 91 71 µ 81 α 1 161 α 1 ( < and Max ( ( 18( 1 ( 1 3 G = G = α α µ Combining the cases (ii-b and (iii-a, (314 follows Sub-case (b µ 7( 1 α ( α( α 16 1 1 3 An easy calculation shows that G( is maximum at = ( ( ( ( ( ( 16µ 1 α 1 7 1 α 8µ 1 α 1 9 1 α Substituting the corresonding value of G( along with the value of C ( α in (3, we obtain (315 Remark 3 Putting A=1 and B= -1 in the theorem we get the estimates for the class R ( α Remark 33 Letting A = 1, B = 1 and α = in the theorem, corollary 3 follows
SECOND HANKEL DETERMINANT 13 / 13 REFERENCES [1] RMGoel and BSMehrok, A subclass of univalent functions, Houston J Math, 8(198, 343-357 [] RMGoel and BSMehrok, A subclass of univalent functions, JAustralMathSoc (Series A, 35(1983, 1-17 [3] WK Hayman, Multivalent functions, Cambridge Tracts in math and math-hysics, No 48 Cambridge university ress, Cambridge (1958 [4] AJanteng, SAHalim and MDarus, Estimates on the second Hankel functional for functions whose derivative has a ositive real art, J Quality Measurement and analysis, JQMA(1 (8 189-195 [5] RJLibera and EJZlotkiewics, Early coefficients of the inverse of a regular convex function, Proceeding Amer MathSoc, 85(198, 5-3 [6] JE Littlewood, On inequalties in the theory of functions Procc London Math Soc 3 (195,481-519 [7] TH MacGregor, The radius of univalence of certain analytic functions, Proc Amer Math Soc 14 (1963 514-5 [8] B S Mehrok Two subclasses of certain analytic functions (to aear [9] J W Noonan and D K Thomas, Coefficient differences and Hankel Determinant of really -valent functions, Proc London Math Soc[ 3],5 (197, 53-54 [1] J W Noonan and D K Thomas, On the second Hankel determinant of a really mean -valent functions Trans American Math Society (3 1976, 337-346 [11] KNoshiro, On the theory of schlicht functions,j Fac Soc Hokkaido univser1, (1934-1935, 19-155 [1] N N Pascu, Alha- starlike functions, Bull Uni Brasov CXIX, 1977 [13] C H Pommerenke, On the coefficients and Hankel determinant of univalent functions, J London math Soc 41, 1966, 111-1 [14] C H Pommerenke, On the Hankel determinant of univalent functions, Mathematica 14 (C, 1967, 18-11 [15] CH Pommerenke, Univalent functions wadenheoch and Rurech, Gotingen, 1975 [16] WW Rogosinski, On coefficients of subordinate functions, Proc London Math Zeith (193, 9-13 [17] M O Reade, On close to convex univalent functions, Michhigan Math J 1955, 59-6 [18] SSWarschawski, On the higher derivative at the boundary in conformal maing, Tran Amer MathSoc, 38(1935, 31-34 M S SAROA, MAHARISHI MARKANDESHAWAR UNIVERSITY, MULLANA GURMEET SINGH (CORRESPONDING AUTHOR, KHALSA COLLEGE, PATIALA GAGANDEEP SINGH, AMRITSAR