ON ESTIMATE FOR SECOND AND THIRD COEFFICIENTS FOR A NEW SUBCLASS OF ANALYTIC AND BI-UNIVALENT FUNCTIONS BY USING DIFFERENTIAL OPERATORS

Transcription

1 Palestine Journal of Mathematics Vol. 8(1)(019), Palestine Polytechnic University-PPU 019 ON ESTIMATE FOR SECOND AND THIRD COEFFICIENTS FOR A NEW SUBCLASS OF ANALYTIC AND BI-UNIVALENT FUNCTIONS BY USING DIFFERENTIAL OPERATORS Ahmad Zireh Saideh Hajiparvaneh Communicated by H. M. Srivastava MSC 010 Classifications: Primary 30C45, Secondary 30C50. Keyords phrases: Analytic functions, Bi-univalent functions, Coefficient estimates, Sălăgean s differential operator. Abstract. In this ork, e firstly introduce a ne subclass B h,p (n, λ) of analytic biunivalent functions. Moreover, e estimate the second third coefficients for functions in this subclass. Our results presented in this paper improve some recent orks. 1 Introduction We denote A for a class of functions of the form f() = + a n n, (1.1) n= hich are analytic in the open unit disk U = { C : < 1}. Assume that S be the class of functions f A hich are univalent in U. From Koebe one-quarter theorem [5], every function f S has an inverse f 1, hich is defined by here f(f 1 ()) = f 1 (f()) = ( U), ( < r 0 (f); r 0 (f) 1 ), 4 g() := f 1 () = a + (a a 3 ) 3 (5a 3 5a a 3 + a 4 ) 4 +. (1.) If f f 1 are univalent in U, then f is bi-univalent. We denote for the class of biunivalent functions in U given by (1.1). There interest to study the bi-univalent functions class (see [6, 19, 0]) derive non-sharp estimates on the first to Taylor-Maclaurin coefficients a a 3. The coefficient estimate problem i.e. bound for a n (n N {, 3}) for each f, is still an open problem. Recently Frasin Aouf [6] investigated the folloing to subclasses of the bi-univalent function class obtained bounds for the first to Taylor-Maclaurin coefficients a a 3 of functions in each of these subclasses. Definition 1.1. (see [6]) Let 0 < α 1 0 λ 1. A function f() is said to be in the class B (α, λ) if the folloing conditions are satisfied: f arg[(1 λ)f() + λf ()] < απ ( U), here the function g is given by (1.). arg[(1 λ)g() + λg ()] < απ ( U),

2 ON ESTIMATE FOR SECOND AND THIRD COEFFICIENTS 45 Theorem 1.. (see [6]) If f() is in the class B (α, λ), then a α (λ + 1) + α(1 + λ λ ), a 3 4α (1 + λ) + α (1 + λ). Definition 1.3. (see [6]) Let 0 β < 1 0 λ 1. We say that a function f() is in the class B (β, λ) if the folloing conditions are satisfied: f Re[(1 λ) f() here the function g is given by (1.). + λf ()] > β ( U), Re[(1 λ) g() + λg ()] > β ( U), Theorem 1.4. (see [6]) If f() is in the class B (β, λ), then (1 β) a λ + 1, a 4(1 β) (1 β) 3 + (1 + λ) (1 + λ). In 1983, Salagean [8] defined differential operator D k : A A as D 0 f() = f(), D 1 f() = Df() = f (), We note that D k f() = D(D k 1 f()) = (D k 1 f()), k N = {1,, 3, }. D k f() = + n k a n n, k N 0 = {0} N. (1.3) n= By using Salagean differential operator, Poral Darus [7] introduced the folloing to subclasses of the bi-univalent function. Definition 1.5. (see [7]) Let n N 0, 0 < α 1 λ 1. A function f() is called in the class B (n, α, λ) if the folloing conditions are satisfied: [ f (1 λ)d n arg f() + λd n+1 ] f() < απ ( U), here the function g is given by (1.). [ (1 λ)d n arg g() + λd n+1 ] g() < απ Theorem 1.6. (see [7]) If f() is in the class B (n, α, λ), then a α 4n (1 + λ) + α[.3 n (1 + λ) 4 n (1 + λ) ], ( U), α [(1 λ)3 n + λ3 n+1 ] + 4α [(1 λ) n + λ n+1 ].

3 46 Ahmad Zireh Saideh Hajiparvaneh Definition 1.7. (see [7]) Let n N 0, 0 β < 1 λ 1. Under the folloing conditions, a function f() is said to be in the class B (n, β, λ): f Re n f() + λd n+1 ) f() > β ( U), Re n g() + λd n+1 ) g() > β ( U), here the function g is given by (1.). Theorem 1.8. (see [7]) If f() is in the class B (n, β, λ), then (1 β) a [(1 λ)3 n + λ3 n+1 ], (1 β) [(1 λ)3 n + λ3 n+1 ] + 4(1 β) [(1 λ) n + λ n+1 ]. The aim of this paper is to investigate the bi-univalent function class B h,p (n, λ) introduced in Definition.1 derive coefficient estimates on the first to Taylor-Maclaurin coefficient a a 3. Our results for the bi-univalent function class f B h,p (n, λ) ould generalie improve some recent orks Srivastava [9], Frasin Aouf [6] Poral Darus [7]. The subclass B h,p (n, λ) In this section, the general subclass B h,p (n, λ) is introduced investigated. Definition.1. Assume that the functions h, p : U C be so constrained that min{re(h()), Re(p())} > 0 ( U) h(0) = p(0) = 1. Let n N 0 λ 1. We say that a function f A given by (1.1) is in the class B h,p (n, λ) if the folloing conditions are satisfied: f n f() + λd n+1 ) f() h(u) ( U), (.1) n g() + λd n+1 ) g() p(u) ( U), (.) here the function g is defined by (1.). Remark.. There are many selections of h p hich ould provide interesting subclasses of class B h,p (n, λ). For example, if e set h() = p() = ( ) α 1 + (0 < α 1, λ 1, U), 1 it can be easily verified that to functions h() p() satisfy the hypotheses of Definition.1. If e have f B h,p (n, λ), then [ f (1 λ)d n arg f() + λd n+1 ] f() < απ ( U),

4 ON ESTIMATE FOR SECOND AND THIRD COEFFICIENTS 47 [ (1 λ)d n arg g() + λd n+1 ] g() < απ ( U), here the function g is given by (1.). Therefore, in this case the class B h,p (n, λ) reduce to class in Definition 1.5 if e take n = 0, it decrease to class in Definition 1.1. If e put h() = p() = 1 + (1 β) 1 (0 β < 1, λ 1, U), then the conditions of Definition.1 are satisfied for both functions h() p(). If f B h,p (n, λ), then f Re n f() + λd n+1 ) f() > β ( U), Re n g() + λd n+1 ) g() > β ( U), here the function g is defined by (1.). Therefore, in this case the class B h,p (n, λ) reduce to class in Definition 1.6 if e set n = 0, it decrease to class in Definition Coefficient Estimates We are no ready to express the bounds for the coefficients a a 3 for subclass B h,p (n, λ). Theorem.3. If f() be in the class B h,p (n, λ), then h a min (0) + p (0) [(1 λ) n + λ n+1 ], h (0) + p (0) 4[(1 λ)3 n + λ3 n+1 ], (.3) min { h (0) + p (0) 4[(1 λ)3 n + λ3 n+1 ] + h (0) + p (0) [(1 λ) n + λ n+1 ], h } (0) [(1 λ)3 n + λ3 n+1. (.4) ] Proof. The main idea of the proof is to get the desired bounds for the coefficient a a 3. Consider relations (.1) (.). No, e have: n f() + λd n+1 ) f() = h() (λ 1, U), (.5) n g() + λd n+1 ) g() = p() (λ 1, U), (.6) here to functions h p satisfy the conditions of Definition.1, respectively. With respect to the folloing Taylor-Maclaurin series expansions for the functions h p, e get: h() = 1 + h 1 + h + h 3 3 +, (.7)

5 48 Ahmad Zireh Saideh Hajiparvaneh p() = 1 + p 1 + p + p (.8) Substituting relations (.7) (.8) into (.5) (.6), respectively, yield Comparing the coefficients (.9) (.11), e obtain [(1 λ) n + λ n+1 ]a = h 1, (.9) [(1 λ)3 n + λ3 n+1 ]a 3 = h, (.10) [(1 λ) n + λ n+1 ]a = p 1, (.11) [(1 λ)3 n + λ3 n+1 ](a a 3 ) = p. (.1) h 1 = p 1, (.13) [(1 λ) n + λ n+1 ] a = h 1 + p 1. (.14) Adding (.10) (.1) give us the folloing relation: [(1 λ)3 n + λ3 n+1 ]a = p + h. (.15) Therefore, considering relations (.14) (.15), e have: a = a = h 1 + p 1 [(1 λ) n + λ n+1 ], (.16) p + h [(1 λ)3 n + λ3 n+1 ], (.17) respectively. So, e find from the equations (.16) (.17), that a h (0) + p (0) [(1 λ) n + λ n+1 ], a h (0) + p (0) 4[(1 λ)3 n + λ3 n+1 ], respectively. Hence, e derive the desired bound on the coefficient a as asserted in (.3). The proof is completed by finding the bound for the coefficient a 3. Subtracting (.1) from (.10) yields [(1 λ)3 n + λ3 n+1 ]a 3 [(1 λ)3 n + λ3 n+1 ]a = h p. (.18) Put the value of a from (.16) into (.18). So, it follos that a 3 = Therefore, e conclude the folloing bound: h 1 + p 1 [(1 λ) n + λ n+1 ] + h p [(1 λ)3 n + λ3 n+1 ]. h (0) + p (0) [(1 λ) n + λ n+1 ] + h (0) + p (0) 4[(1 λ)3 n + λ3 n+1 ]. (.19)

6 ON ESTIMATE FOR SECOND AND THIRD COEFFICIENTS 49 By taking the value of a from (.17) into (.18), it follos that a 3 = Hence, e get: (p + h ) [(1 λ)3 n + λ3 n+1 ] + (h p ) [(1 λ)3 n + λ3 n+1 ] = h [(1 λ)3 n + λ3 n+1 ]. h (0) [(1 λ)3 n + λ3 n+1 ]. (.0) No, e obtain from (.19) (.0) the desired estimate for the coefficient a 3 as asserted in (.4) the proof is completed. 3 Conclusions If e set h() = p() = in Theorem.3, then Corollary 3.1 can be obtained. ( ) α 1 + (0 < α 1, U), 1 Corollary 3.1. Let the function f() given by (1.1) be in the class B (n, α, λ). Then { } α a min [(1 λ) n + λ n+1 ], [(1 λ)3 n + λ3 n+1 ] α, α [(1 λ)3 n + λ3 n+1 ]. Remark 3.. For the coefficient a 3, it is easily seen that α [(1 λ)3 n + λ3 n+1 ] α [(1 λ)3 n + λ3 n+1 ] + 4α [(1 λ) n + λ n+1 ]. So, it is clear that Corollary 3.1 is an improvement of Theorem 1.6. Set n = 0 in Corollary 3.1. So, e have the folloing corollary. Corollary 3.3. Let the function f() given by (1.1) be in the class B (α, λ). Then a λ + 1 α, 1 λ < 1 + α λ + 1, λ 1 + α λ + 1. Remark 3.4. Corollary 3.3 improves Theorem 1.. If e take λ = 1 in Corollary 3.3, then e get the folloing corollary.

7 50 Ahmad Zireh Saideh Hajiparvaneh Corollary 3.5. Let the function f() given by (1.1) be in the class H α (0 < α 1). Then a 3 α, α 3. Remark 3.6. Corollary 3.5 provides a refinement of a result hich obtained by Srivastava [9, Theorem 1]. Put h() = p() = 1 + (1 β) 1 in Theorem.3. Then, Corollary 3.7 can be easily concluded. (0 β < 1, U), Corollary 3.7. Let the function f() given by (1.1) be in the class B (n, β, λ). Then { } (1 β) a min [(1 λ) n + λ n+1 ], (1 β) [(1 λ)3 n + λ3 n+1, ] (1 β) [(1 λ)3 n + λ3 n+1 ]. Remark 3.8. For the coefficient a 3, it can be easily seen that (1 β) [(1 λ)3 n + λ3 n+1 ] (1 β) [(1 λ)3 n + λ3 n+1 ] + 4(1 β) [(1 λ) n + λ n+1 ]. (3.1) Hence, Corollary 3.7 improves Theorem 1.8 ith regard to relation (3.1). By setting n = 0 in Corollary 3.7, e get: Corollary 3.9. Let the function f() given by (1.1) be in the class B (β, λ) here λ 1. Then (1 β) a min λ + 1, (1 β) λ + 1, (1 β) λ + 1. Remark Theorem 1.4 is improved by Corollary 3.9. If e put λ = 1 in Corollary 3.9, the folloing corollary can be obtained. Corollary Let the function f() given by (1.1) be in the class H (β)(0 β < 1). Then (1 β), 0 β a 1 (1 β), 3 β < 1 (1 β). 3 Remark 3.1. The bounds on a a 3 given in Corollary 3.11 are better than those given by Srivastava [9, Theorem ].

8 ON ESTIMATE FOR SECOND AND THIRD COEFFICIENTS 51 References [1] S. Bulut, Coefficient estimates for a class of analytic bi-univalent functions, Novi Sad J. Math. 43(), (013). [] S. Bulut, Coefficient Estimates for Ne Subclasses of Analytic Bi-Univalent Functions Defined by Al-Oboudi Differential Operator, J. Funct. Spaces Appl., Article ID 18193, 7 pages (013). [3] M. Caglar, H. Orhan N. Yagmur, Coefficient bounds for ne subclasses of bi-univalent functions, Filomat 7(7), (013). [4] M. Caglar, E. Deni H. M. Srivastava, Second Hankel determinant for certain subclasses of biunivalent functions, Turkish J. Math. 41, (017). [5] P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, B 59, Springer- Verlag, Ne York, Berlin, Heidelberg Tokyo, [6] B. A. Frasin M. K. Aouf, Ne subclasses of bi-univalent functions, Appl. Math. Lett. 4, (011). [7] S. Poral M. Darus, On a ne subclass of bi-univalent functions, J. Egyptian Math. Soc. 1, (013). [8] G. S. Sălăgean, Subclasses of univalent functions, in Complex Analysis Fifth Romanian Finish Seminar, Part 1 (Bucharest, 1981) 1013 of Lecture Notes in Mathematics, Springer, Berlin, Germany, (1983). [9] H. M. Srivastava, A. K. Mishra P. Gochhayat, Certain subclasses of analytic bi-univalent functions, Appl. Math. Lett. 3, (010). [10] H. M. Srivastava, S. Bulut, M. Caglar N. Yagmur, Coefficient estimates for a general subclass of analytic bi-univalent functions, Filomat 7(5), (013). [11] H. M. Srivastava, S. Gaboury F. Ghanim, Coefficient estimates for some general subclasses of analytic bi-univalent functions, Afr. Mat. 8, (017). [1] H. M. Srivastava, S. Gaboury F. Ghanim, Initial coefficient estimates for some subclasses of m-fold symmetric bi-univalent functions, Acta Math. Sci. Ser. B Engl. Ed. 36, (016). [13] H. M. Srivastava, B. S. Joshi, S. Joshi H. Paar, Coefficient estimates for certain subclasses of meromorphically bi-univalent functions, Palest. J. Math. 5, Special Issue, (016). [14] H. M. Srivastava, S. Sumer Eker M. Rosihan Ali, Coefficient bounds for a certain class of analytic bi-univalent functions, Filomat 9, (015). [15] H. M. Srivastava D. Bansal, Coefficient estimates for a subclass of analytic bi-univalent functions, J. Egyptian Math. Soc. 3, 4 46 (015). [16] H. M. Srivastava, S. Gaboury F. Ghanim, Coefficient estimates for some subclasses of M-fold symmetric bi-univalent functions, Acta Univ. Apulensis Math. Inform. 3, (015). [17] H. M. Srivastava, S. Sivasubramanian R. Sivakumar, Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions, Tbilisi Math. J. 7, 1 10 (014). [18] Huo Tang, H. M. Srivastava, S. Sivasubramanian P. Gurusamy, The Fekete-Sego functional problems for some subclasses of m-fold symmetric bi-univalent functions, J. Math. Inequal. 10, (016). [19] Q.- H. Xu, Y.- C. Gui H. M. Srivastava, Coefficient estimates for a Certain subclass of analytic bi-univalent functions, Appl. Math. Lett. 5, (01). [0] Q. -H. Xu, H. -G. Xiao H. M. Srivastava, A certain general subclass of analytic bi-univalent functions associated coefficient estimate problems, Appl. Math. Comput. 18(3), (01). Author information Ahmad Zireh Saideh Hajiparvaneh, Faculty of Mathematicsal Sciences, Shahrood University of Technology, Iran. aireh@shahroodut.ac.ir Received: June 7, 017. Accepted: September 1, 017