ESTIMATE FOR INITIAL MACLAURIN COEFFICIENTS OF CERTAIN SUBCLASSES OF BI-UNIVALENT FUNCTIONS

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1 Miskolc Matheatical Notes HU e-issn Vol. 7 (07), No., pp DOI: 0.854/MMN ESTIMATE FOR INITIAL MACLAURIN COEFFICIENTS OF CERTAIN SUBCLASSES OF BI-UNIVALENT FUNCTIONS BADR S. ALKAHTANI, RANAY GOSWAMI, AND TEODOR BULBOACĂ Received 6 February, 05 Abstract. In this paper, estiates for second third MacLaurin coefficients of certain subclasses of bi-univalent functions in the open unit disk defined by convolution are deterined, certain special cases are also indicated. The ain result extends iprove a recent one obtained by Srivastava et al. 00 Matheatics Subject Classification: 30C45 30C50 Keywords: univalent functions, bi-univalent functions, bi-starlike function, bi-convex function, functions with bounded boundary rotation, coefficient estiates, convolution (Hadaard) product. INTRODUCTION AND DEFINITIONS Let A be the class of functions f of the for f. / D C a n n, which are analytic in the open unit disk D D f C W j j < g noralized by the conditions f.0/ D 0 f 0.0/ D. The Koebe one-quarter theore [3] ensures that the iage of D under every univalent function f A contains the disk with the center in the origin the radius =4. Thus, every univalent function f A has an inverse f W f.d/! D, satisfying f.f. // D, D, f f.w/ D w jwj < r 0.f / r 0.f / 4 : Moreover, it is easy to see that the inverse function has the series expansion of the for f.w/ D w a w C a a 3 w 3 5a 3 5a a 3 C a 4 w 4 C ::: w f.d/: (.) A function f A is said to be bi-univalent, if both f f are univalent in D, in the sense that f has a univalent analytic continuation to D, we denote by this class of bi-univalent functions. c 07 Miskolc University ress

2 740 BADR S. ALKAHTANI, RANAY GOSWAMI, AND TEODOR BULBOACĂ In [8] the authors defined the classes of functions./ as follows: let./, with 0 <, denote the class of univalent analytic functions, noralized with.0/ D, satisfying Z Re. / d 0 where D re i D. For D 0, we denote WD.0/, hence the class represents the class of functions p analytic in D, noralized with p.0/ D, having the representation Z p. / D 0 e it d.t/ (.) C eit where is a real-valued function with bounded variation, which satisfies Z 0 d.t/ D Z 0 jd.t/j : (.3) Clearly, WD is the well-known class of Carathéodory functions, i.e. the noralized functions with positive real part in the open unit disk D. Lewin [6] investigated the class of bi-univalent functions obtained the bound for the second coefficient. Brannan Taha [] considered certain subclasses of biunivalent functions, siilar to the failiar subclasses of univalent functions consisting of strongly starlike, starlike convex functions. They introduced the concept of bi-starlike functions the bi-convex functions, obtained estiates for the initial coefficients. Recently, Ali et al. [], Srivastava et al. [9], Frasin Aouf [4], Goyal Goswai [5] any others have introduced investigated subclasses of bi-univalent functions obtained bounds for the initial coefficients. Motivated by work of Srivastava et al. [9], we introduce a new subclass of bi-univalent functions, as follows. For the functions f h A given by X X f. / D C a n n h. / D C b n n D we recall the Hadaard (or convolution) product of f h, defined by X.f h/. / D C a n b n n D: Definition. For a given function k, a function f is said to be in the class BR k.i/, with 0 <, if the following conditions are satisfied.f k/. /./

3 ESTIMATE FOR INITIAL MACLAURIN COEFFICIENTS 74 where g D f w D..g k/.w/ w./ Reark. Taking k. / D =. / D in the Definition we obtain the class B./ WD BR =. /.I/ studied by Srivastava et al. [9, Definition ]. Definition. For a given function k a nuber C, a function f is said to be in the class BV k.i /, with 0 <, if the following conditions are satisfied where g D f.. /.f k/0. /.f k/. / C. / w.g k/0.w/.g k/.w/ C.f k/00. /.f k/ 0./. / C C w.g k/00.w/.g k/ 0./.w/ Reark..i/ Taking D 0 D in the above class BV k.i / we obtain the classes S k./ WD BV k.i0/ C k./ WD BV k.i/, respectively..ii/ Moreover, if we take k. / D =. / D, the classes S k./ C k./ reduces to the well-known classes of bi-starlike bi-convex functions, respectively (see also []). The object of the paper is to find estiates for the coefficients a a 3 for functions in the subclass BR k.i/ BV k.i /, these bounds are obtained by eploying the techniques used earlier by Srivastava et al. [9].. MAIN RESULTS In order to prove our ain result for the functions f BR k.i/, first we will prove the following lea: Lea. Let the function. / D C h n n, D, such that./. Then, nd jh n j n : roof. Fro (.) (.3), like in [8] [7], we can see that if p, then p. / D 4 C p. / p. / (.) 4 where p p.

4 74 BADR S. ALKAHTANI, RANAY GOSWAMI, AND TEODOR BULBOACĂ Further, if p. / D C p n n, D, where p. / D C p. / D C nd nd nd p n./ n p n./ n for all D, coparing the coefficients of both sides of (.) we get p n D 4 C p n./ p n./ 4 n : Since p p, where is the class of Carathéodory functions, it is well-known that jp n./ j jp n./ j for all n, thus jp n j 4 C././ p n C p n 4 4 C C D n : (.) 4 Now, the proof of this lea is straight forward, if we write X. / D. /p. / C where p. / D C p n n : Then, which gives. / D C. / nd X p n n D nd h n D. /p n n using the inequality (.) we obtain the desired result. Theore. Let f. / D C a n n be in the class BR k.i/, where k has the for k. / D C k n n. If k k 3 0, then ja j in (s I ) ja 3 j jk j a a 3 : roof. Since f BR k.i/, fro the Definition we have.f k/. / D p. / (.3)

5 ESTIMATE FOR INITIAL MACLAURIN COEFFICIENTS 743.g k/.w/ D q.w/ (.4) w where p q./ g D f. Using the fact that the functions p q have the following Taylor expansions p. / D C p C p C p 3 3 C ::: D (.5) q.w/ D C q w C q w C q 3 w 3 C ::: w D (.6) equating the coefficients in (.3) (.4), fro (.) we get k a D p (.7) k 3 a 3 D p (.8) k a D q k 3 a a 3 D q : (.9) Since p q./, according to Lea, the next inequalities hold: jp k j k (.0) jq k j k (.) thus, fro (.8) (.9), by using the inequalities (.0) (.), we obtain which gives ja j jq j C jp j ja j s Fro (.7), by using (.0) we obtain iediately that ja j D p jk j k : (.) cobining this with the inequality (.), the first inequality of the conclusion is proved. According to (.8), fro (.0) we easily obtain ja 3 j D p k 3 fro (.9), by using (.0) (.) we finally deduce a a 3 D q which copletes our proof. k 3

6 744 BADR S. ALKAHTANI, RANAY GOSWAMI, AND TEODOR BULBOACĂ Setting D 0 in Theore we get the following special case: Corollary. Let f. / D C a n n be in the class BR k.i0/, where k has the for k. / D C k n n. If k k 3 0, then r ja j in I For k. / D =. jk j ja 3 j a a 3 : / the above corollary reduces to the next result: Exaple. If f. / D C a n n is in the class BR =. /.I0/, then ja j Taking k. / D =. r 3 ja 3j 3 a a 3 3 : / in Corollary, we get: Exaple. If f. / D C a n n is in the class BR =. /.I0/, then If we put k. / D =. ja j p ja 3 j a a 3 : / in Theore, we deduce the next corollary: Corollary. If f. / D C a n n is in the class B./, then q. / 3 if < ja j : if ja 3 j 3 < <. / a a 3 : 3. / 3 Reark 3. For the special case 3 < <, the above first inequality, the second one for all 0 <, iprove the estiates given by Srivastava et al. in [9, Theore ]. Theore. Let f. / D C a n n be in the class BV k.i /, with C n f g, where k has the for k. / D C k n n. If k k 3 0. C /k 3. C 3 /k 0

7 then ESTIMATE FOR INITIAL MACLAURIN COEFFICIENTS 745 (s ) ja j in. C /k3. C 3 /k I j C jjk j ( ja 3 j in. C /k3. C 3 /k C j C j I j C 3 j C j C j j C j I C 4.!) C /k3. C 3 /k j C j jk j j C j whenever C n. roof. If f BV k.i /, according to the Definition we have. /.f k/0. /.f k/. / C C.f k/00. /.f k/ 0 D p. /. /. / w.g k/0.w/ C C w.g k/00.w/.g k/.w/.g k/ 0.w/ where p q./ g D f. Since. /.f k/0. /.f k/. / C C.f k/00. /.f k/ 0. / D D q.w/ C. C /a k C. C /a 3 k 3. C 3 /a k C ::: D according to (.). /.g k/0.w/.g k/.w/ C C.g k/00.w/.g k/ 0 D. C /a k wc.w/ n 4. o C /k3. C 3 /k a. C /a 3 k 3 w C ::: w D fro (.5) (.6) cobined with the above two expansion forulas, it follows that. C /a k D p (.3). C /a 3 k 3. C 3 /a k D p (.4). C /a k D q 4. C /k3. C 3 /k a. C /a 3 k 3 D q : (.5)

8 746 BADR S. ALKAHTANI, RANAY GOSWAMI, AND TEODOR BULBOACĂ Now, fro (.4) (.5) we deduce that whenever. C /k 3 a D p C q 4. C /k 3. C 3 /k (.6). C 3 /k 0, 4. C /k 3 a 3 a D p q : Using (.6) in the above relation, we obtain p C q a 3 D 4. C /k 3. C 3 /k C p q (.7) 4. C /k 3 whenever. C /k 3. C 3 /k 0 C n : Fro (.3) (.4) we get a 3 D p C C 3. C /k 3. C / p for C n I (.8) while fro (.3) (.5) we deduce that " a 3 D q C 4. C /k # 3. C 3 /k. C /k 3 k. C p / (.9) for C n I. Cobining (.3) (.6) for the coputation of the upper-bound of ja j, (.7), (.8) (.9) for the coputation of ja 3 j, by using Lea we easily find the estiates of our theore. Taking D 0 D in Theore we obtain the following two special cases, respectively: Corollary 3. Let f. / D C the for k. / D C k n n. If k k 3 0 then a n n be in the class S k./, where k has k 3 k 0 (s ja j in k3 k I ( ja 3 j in k3 k C I ) jk j. C / I

9 ESTIMATE FOR INITIAL MACLAURIN COEFFICIENTS 747 Corollary 4. Let f. / D C the for k. / D C k n n. If k k 3 0 then C.!) /4k3 k jk j : a n n be in the class C k./, where k has 3k 3 k 0 (s ja j in 6k3 4k I ( ja 3 j in 6k3 4k C I 6 6 ACKNOWLEDGEMENT ) jk j. C / I 6 C.!) /3k3 k jk j : Dr. Badr S. Alkahtani Dr. ranay Goswai extend their sincere appropriations to the Deanship of Scientific Research at King Saud University for its funding this rofile Research Group (RG ). REFERENCES [] R. M. Ali, S. K. Lee, V. Ravichran, S. Supraania, Coefficient estiates for bi-univalent Ma-Minda stalike convex functions, Appl. Math. Lett., vol. 5, no. 3, pp , 0, doi: 0.06/j.al [] D. A. Brannan T. S. Taha, On soe classes of bi-univalent functions, Stud. Univ. Babeş- Bolyai Math., vol. 3, no., pp , 986. [3]. L. Duren, Univalent functions, in Grundlehren der Matheatischen Wissenschaften Series. New York: Springer Verlag, 983, vol. 59. [4] B. A. Frasin M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., vol. 4, no. 9, pp , 0, doi: 0.06/j.al [5] S.. Goyal. Goswai, Estiate for initial Maclaurin coefficients of bi-univalent functions for a class defined by fractional derivatives, J. Egyptian Math. Soc., vol. 0, no. 3, pp. 79 8, 0, doi: 0.06/j.joes [6] M. Lewin, On a coefficient proble for bi-univalent functions, roc. Aer. Math. Soc., vol. 8, pp , 967, doi: 0.307/0355. [7] K. I. Noor, W. Ul-Haq, M. Arif, S. Mustafa, On bounded boundary bounded radius rotations, J. Inequal. Appl., vol. 009, article ID: 83687, pages, doi: 0.55/009/ [8] K. adanabhan R. arvatha, roperties of a class of functions with bounded boundary rotation, Ann. olon. Math., vol. 3, pp. 3 33, 975.

10 748 BADR S. ALKAHTANI, RANAY GOSWAMI, AND TEODOR BULBOACĂ [9] H. M. Srivastava, A. K. Mishra,. Gochhayat, Certain subclasses of analytic bi-univalent functions, Appl. Math. Lett., vol. 3, no. 0, pp. 88 9, 00, doi: 0.06/j.al Authors addresses Badr S. Alkahtani King Saud University, Matheatics Departent, College of Science,. O. Box 4, Riyadh 989, Saudi Arabia E-ail address: ranay Goswai Abedkar University Delhi, School of Liberal Studies, Delhi-0006, India E-ail address: Teodor Bulboacă Babeş-Bolyai University, Faculty of Matheatics Coputer Science, Cluj-Napoca, Roania E-ail address: