Some properties for α-starlike functions with respect to k-symmetric points of complex order

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1 doi: /a A N N A L E S U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N P O L O N I A VOL. LXXI, NO. 1, 217 SECTIO A 1 9 H. E. DARWISH, A. Y. LASHIN and S. M. SOWILEH Some properties for α-starlie functions with respect to -symmetric points of complex order Abstract. In the present wor, we introduce the subclass Tγ,α(ϕ, of starlie functions with respect to -symmetric points of complex order γ (γ in the open unit disc. Some interesting subordination criteria, inclusion relations and the integral representation for functions belonging to this class are provided. The results obtained generalize some nown results, and some other new results are obtained. 1. Introduction. Let A be the class of functions f of the form: (1.1 f = z + a z, which are analytic in the open unit dis = {z : z C, z < 1}. A function f A is subordinate to an univalent function g A, written f g, if f( = g( and f( g(. Let Ω be the family of analytic functions w in the unit disc satisfying the conditions w( = and w < 1, for z. Note that f g if there is a function w Ω such that f = g(w. Further, let be the class of analytic functions p with p( = 1, which are convex and univalent in and satisfy =2 21 Mathematics Subject Classification. 3C45. Key words and phrases. Analytic functions, starlie functions of complex order, convex functions of complex order, α-starlie, -symmetric points, subordination.

2 2 H. E. Darwish, A. Y. Lashin and S. M. Sowileh the condition Re {p} >, (z. For a given positive integer, let ε = exp(2πi/ and (1.2 f = 1 1 ε v f(ε v z, v= (z. Let ϕ, we denote by Ss (ϕ, Cs (ϕ and Tα (ϕ the familiar subclasses of A consisting of starlie, convex and α-starlie functions with respect to -symmetric points in respectively. That is { Ss (ϕ = f A : zf } ϕ (z, and (1.3 T α (ϕ = f Cs (ϕ = {f A : (zf f } ϕ (z, { f A : αz (zf + (1 αzf } αzf + (1 αf ϕ (z. The class Tα (ϕ was introduced and studied by Parvatham and Radha [11] and this class generalizes the classes defined by Pascu [12], Das and Singh [5] as well as the classes Ss (ϕ and Cs (ϕ which were studied recently by Wang et al. [18]. Recently several subclasses of analytic functions with respect to -symmetric points were introduced and studied by various authors (see [1], [2], [14], [18], [19], [21]. Following Ma and Minda [7], Ravichandran et al. [15] defined a more general class related to the class of starlie functions of complex order as follows. A function f A is said to be in the class S γ (ϕ, if it satisfies the following subordination condition ( zf γ f 1 ϕ (z, γ C = C \ {} and ϕ. Furthermore, a function f A is said to be in the class C γ (ϕ, if it also satisfies the subordination condition ( zf γ f ϕ (z, γ C = C \ {} and ϕ. Motivated by the classes T α (ϕ, S γ (ϕ and C γ (ϕ, we now introduce and investigate the following subclasses of A, and obtain some interesting results. Moreover, for some non-zero complex number γ, we consider the subclasses T γ,α(ϕ, F γ,α(ϕ of A as follows.

3 Some properties for α-starlie functions... 3 Definition 1. Let Tγ,α(ϕ denote the class of functions f in A satisfying the following condition ( γ where ϕ and α. ( αz (zf + (1 αzf αzf + (1 αf 1 ϕ, Definition 2. Let Fγ,α(ϕ denote the class of functions f A satisfying the subordination condition ( ( αz (zf + (1 αzf γ αzξ + (1 αξ 1 ϕ, where ξ = 1 1 v= ε v ξ(ε v z, ξ Tγ,α(ϕ, ϕ, and α. By giving specific values to the parameters, γ and α in the class T γ,α(ϕ, we get the following new subclass of analytic functions. Remar 1. Putting = 1, we obtain the following definition. Definition 3. Let T γ,α (ϕ denote the class of functions f A satisfying the following condition ( ( αz (zf + (1 αzf γ αzf 1 ϕ, + (1 αf where ϕ and α. Remar 2. If we set α =, we have the following definition. Definition 4. Let Sγ (ϕ denote the class of functions f in A satisfying the following condition ( ( zf γ f 1 ϕ, where ϕ. Remar 3. (i T1,α (ϕ = K (α, ϕ (see Paravatham and Radha [11], where K (α, ϕ is the class of α-starlie functions with respect to symmetric points. (ii Tγ,α 1 = SC(γ, α, β ( β < 1 (see Altintas et al. [3]. ( 1+(1 2βz 1 z (iii T1, (ϕ = S s (ϕ and T1,1 (ϕ = C s (ϕ (see Wang et al. [18]. (iv Tγ, 1 (ϕ = S γ(ϕ (see Ravichandran et al. [15]. (v T 1 1+z = K(α (see Pascu and Podaru [13], where K(α is the class of α-starlie functions. (vi T1,1 2 ( 1 z 1+z = C s (see Das and Singh [5], where C s is the class of convex functions with respect to symmetric points. (vii T1, 2 ( 1 z 1+z = S s (see Saaguchi [17], where S s is the class of starlie functions with respect to symmetric points. 1,α ( 1 z

4 4 H. E. Darwish, A. Y. Lashin and S. M. Sowileh (viii T1 β, ( 1 z 1+z = S, s (β and T1 β,1 ( 1 z 1+z = C s (β ( β < 1 (see Chand and Singh [4], where Ss, (β is the class of starlie functions with respect to -symmetrical points of order β and Cs (β is the class of convex functions with respect to -symmetrical points of order β. (viiii Tγ, 1 ( 1 z 1+z = S γ (see Nasr and Aouf [9], where S γ is the class of starlie functions of complex order. (x Tγ,1 1 ( 1 z 1+z = C γ (see Nasr and Aouf [8] and Wiatrowsi [2] where C γ is the class of convex functions of complex order. (xi T1 α, 1 ( 1 z 1+z = S(α and T 1 1 α,1 ( 1 z 1+z = C(α (see Robertson [16], where S(α is the class of starlie functions of order α ( α < 1 and C(α is the class of convex functions of order α ( α < 1. To prove our main results, we need the following lemmas. Lemma 1 ([1]. Let, ϑ be complex numbers. Suppose that h is convex and univalent in with (1.8 h( = 1 and Re [h + ϑ] > (z, and let q be analytic in with q( = 1 and q h. If p = 1 + p 1 z + p 2 z 2 + with p( = 1, then implies that p h. p + zp q + ϑ h Lemma 2 (see [6, 7]. Let, ϑ be complex numbers. Suppose that h is convex and univalent in and satisfies (1.8. If p = 1 + p 1 z + p 2 z and satisfies the subordination then p + zp p + ϑ h, p h. 2. Main result. Unless otherwise mentioned, we assume throughout this article that f A, α >, ϕ and γ C. Proposition 1. Let f T γ,α (ϕ and Re 1 α [αγ (ϕ 1 + 1] >, then f S γ (ϕ. Proof. Set (2.1 p = γ ( zf f 1,

5 Some properties for α-starlie functions... 5 then p is an analytic function with p( = 1. By differentiating (2.1 logarithmically, we get ( αz (zf + (1 αzf γ αzf 1 + (1 αf αzp = p + ϕ, (z. 1 + αγ [p 1] The conclusion of Proposition 1 yields from Lemma 2, by taing = γ and ϑ = 1 αγ α. Similarly, we can prove the following proposition. Proposition 2. Let Re 1 α [αγ (ϕ 1 + 1] >. Then 1 z (2.2 F = I α (f = αz (1/α 1 t (1/α 2 f(tdt S γ (ϕ whenever f S γ (ϕ. Theorem 1. Let f T γ,α(ϕ and Re 1 α [1 + αγ (ϕ 1] >. Then f defined by (1.2 is in T γ,α (ϕ. Further, we have f S γ (ϕ. Proof. Let f Tγ,α(ϕ. Replacing z by ε µ z (µ =, 1,..., 1; ε = 1 in (1.4, then (1.4 also holds true, that is, ( ( (1 αε µ zf (ε µ z + αε µ z [f (ε µ z + ε µ zf (ε µ z] γ (1 αf (ε µ z + αε µ zf 1 ϕ. (εµ z According to the definition of f and ε = 1, we now that (2.4 f (ε µ z = ε µ f and f (εµ z = f = 1 1 f (ε µ z, for any µ =, 1,..., 1, and summing up, we can get 1 1 [ ( (1 αzf (ε µ z + αz [zf (ε µ z] γ (1 αf + αzf (εµ z ] 1 = ( (1 αzf + αz [zf ] γ (1 αf + αzf 1. Hence there exist ζ µs, in such that ( (1 αzf + αz [zf ] γ (1 αf + αzf ϕ(ζ µ = ϕ(ζ, for ζ, since ϕ( is convex. Thus f T γ,α (ϕ.

6 6 H. E. Darwish, A. Y. Lashin and S. M. Sowileh Remar 4. Putting γ = 1 λ ( λ < 1 and ϕ = 1+z 1 z in the above theorem, we get the result obtained by Wang et al. [19, Lemma 3, p. 11]. Theorem 2. Let f Tγ,α(ϕ and Re 1 α [1 + αγ [ϕ 1]] >. Then f Sγ (ϕ. Proof. Let f Tγ,α(ϕ. Then by Definition 1, we have ( αz (zf + (1 αzf γ αzf + (1 αf 1 ϕ. ( zf f 1 and q = γ Putting p = γ easy to obtain that γ ( αz (zf + (1 αzf αzf + (1 αf = p + 1 αzp ϕ, (z. 1 + αγ [q 1] ( zf f 1, it is Since f Tγ,α(ϕ, then by using Theorem 1, we can see that q ϕ. Now an application of Lemma 1, yields p = ( zf γ f 1 ϕ. That is f S γ (ϕ. We thus complete the proof of Theorem 2. Theorem 3. Let f Sγ (ϕ and Re 1 α [1 + αγ (ϕ 1] > in and let F be the integral operator defined by (2.2, then F Sγ (ϕ. Proof. Let a function f of the form (1.2 with F be put in the place of f. That is F = 1 1 v= ε v f(ε v z. We can see that F = ( 1 t (1/α 2 f (t dt and then differentiating with respect to z, we z αz (1/α 1 get (2.5 (1 αf + αzf = f. From (2.2, we have (2.6 (1 αf + αzf = f. Since f Sγ (ϕ, we can apply Theorem 1 with α = to deduce f S γ (ϕ. Applying Proposition 2, we have F S γ (ϕ, that is ( ( zf γ F 1 ϕ. If we denote (2.8 p = γ ( zf F 1,

7 Some properties for α-starlie functions... 7 and (2.9 q = γ ( zf F 1, then p is analytic in with p( = 1 and q is analytic in with q( = 1, q ϕ. Differentiating in (2.6 and using (2.8, we have αγzp (2.1 γp γ ( 1 α + α zf F In view of (2.5 and (2.9, (2.1 gives p + αzp 1 + αγ (q 1 = γ = zf (1 αf + αzf. ( zf f 1 ϕ. Now applying Lemma 1, we get p ϕ, which proves the theorem. The method of proving Theorem 4 is similar to that of Theorem 2. Theorem 4. Let Re 1 α [1 + αγ [ϕ 1]] >. Then F γ,α(ϕ F γ, (ϕ. Now, we give the integral representations of functions belonging to the classes T γ,α(ϕ. Theorem 5. Let f Tγ,α(ϕ, then we have (2.11 f = 1 z α z1 1 γ 1 α exp ε µ t where f is given by equality (1.2 and ω Ω. ϕ(ω(ξ 1 ξ dξ t 1 α 1 dt, Proof. Suppose that f Tγ,α(ϕ. We now that the condition (1.5 can be written as follows ( γ ( αz (zf + (1 αzf αzf + (1 αf 1 = ϕ(ω. By similar application of the arguments given in the proof for Theorem 1 to (2.12, we obtain ( γ From (2.13, we have ( αz (zf + (1 αzf αzf + (1 αf 1 = 1 1 ϕ(ω(ε µ z. (2.14 α (zf + (1 αf αzf + (1 αf 1 z = γ 1 ϕ(ω(ε µ z 1. z

8 8 H. E. Darwish, A. Y. Lashin and S. M. Sowileh Integrating this equality, we get [ αzf ] log + (1 α f = γ 1 z that is, (2.15 αzf + (1 α f = z exp γ or equivalently, (2.16 (1 α f + αzf = z exp γ z 1 z ϕ(ω(ε µ ρ 1 dρ, ρ 1 ε µ z ϕ(ω(ε µ ρ 1 dρ, ρ ϕ(ω(ζ 1 dζ. ζ From (2.16, we can get equality (2.11 easily. This completes the proof of Theorem 5. Remar 5. (i Putting γ = 1 in the above theorem, we get the result obtained by Wang et al. [18, Theorem 5, p. 12]. (ii Putting γ = 1 and α = 1 in the above theorem, we get the result obtained by Wang et al. [18, Theorem 3, p. 11]. (iii For γ = 1 and ϕ = 1+(1 2λz 1 z ( λ < 1 in the above theorem, we get the result obtained by Wang et al. [19, Theorem 1, p. 112]. (iv Putting γ = 1 in our results, we get the results obtained by Paravatham and Radha [11]. References [1] Abubaer, A. A. A., Darus, M., On starlie and convex functions with respect to -symmetric points, Internat. J. Math. Math. Sci., 211 (211, Art. ID 83464, 9 p. [2] Al-Oboudi, F. M., On classes of functions related to starlie functions with respect to symmetric conjugate points defined by a fractional differential operator, Complex Anal. Oper. Theory 5 (211, [3] Altintas, O., Irma, H., Owa, S., Srivastava, H. M., Coefficient bounds for some families of starlie and convex functions of complex order, Appl. Math. Letters 2 (27, [4] Chand, R., Singh, P., On certain schlicht mappings, Indian J. Pure Appl. Math. 1 (9 (1979, [5] Das, R. N., Singh, P., On subclasses of schlicht mapping, Indian J. Pure Appl. Math. 8 (1977, [6] Eenigenburg, P., Miller, S. S., Mocanu, P. T., Reade, M. O., On a Briot Bouquet differential subordination, Rev. Roumaine Math. Pures Appl. 29 (1984, [7] Ma, W. C., Minda, D. A., A unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analysis, Z. Li, F. Ren, L. Yang, and S. Zhang (Eds., Int. Press, 1994, [8] Nasr, M. A., Aouf, M. K., On convex functions of complex order, Mansoura Sci. Bull. Egypt. 9 (1982,

9 Some properties for α-starlie functions... 9 [9] Nasr, M. A., Aouf, M. K., Starlie functions of complex order, J. Natur. Sci. Math. 25 (1985, [1] Padmanabhan, K. S., Parvatham, R., Some applications of differential subordination, Bull. Aust. Math. Soc. 32 (3 (1985, [11] Parvatham, R., Radha, S., On α-starlie and α-close-to-convex functions with respect to n-symmetric points, Indian J. Pure Appl. Math. 16 (9 (1986, [12] Pascu, N. N., Alpha-close-to-convex functions, in: Romanian Finnish Seminar on Complex Analysis (Bucharest, 1976, Springer, Berlin, 1979, [13] Pascu, N. N., Podaru, V., On the radius of alpha-starlieness for starlie functions of order beta, in: Complex analysis fifth Romanian Finnish seminar, Part 1 (Bucharest, 1981, Springer, Berlin, 1983, [14] Ramachandran, C., Kavitha, D., Soupramanien, T., Certain bound for q-starlie and q-convex function with respect to symmetric points, Internat. J. Math. Math. Sci. 215 (215, Art. ID 25682, 7 pp. [15] Ravichandran, V., Polatoglury, Y., Bolcaly, M., Seny, A., Certain subclasses of starlie and convex functions of complex order, Hacettepe J. Math. Stat. 34 (25, [16] Robertson, M. S., On the theory of univalent functions, Ann. Math. 37 (1936, [17] Saaguchi, K., On certain univalent mapping, J. Math. Soc. Japan 11 (1959, [18] Wang, Z.-G., Gao, C.-Y., Yuan, S.-M., On certain subclasses of close-to-convex and quasi-convex functions with respect to -symmetric points, J. Math. Anal. Appl. 322 (1 (26, [19] Wang, Z.-G., Gao, C.-Y., Orhan, H., Abulut, S., Some subclasses of close-to-convex and quasi-convex functions with respect to -symmetric points, General Math., 15 (4 (27, [2] Wiatrowsi, P., The coefficient of a certain family of holomorphic functions, Zeszyty Nau. Uniw. Łódz. Naui Mat. Przyrod. Ser. II No. 39 Mat. (1971, [21] Yuan, S.-M., Liu, Z.-M., Some properties of α-convex and α-quasiconvex functions with respect to n-symmetric points, Appl. Math. Comput. 188 (2 (27, H. E. Darwish A. Y. Lashin Department of Mathematics Department of Mathematics Faculty of Science Faculty of Science Mansoura University Mansoura University Mansoura, Mansoura, Egypt Egypt Darwish333@yahoo.com aylashin@mans.edu.eg S. M. Sowileh Department of Mathematics Faculty of Science Mansoura University Mansoura, Egypt s soileh@yahoo.com Received April 19, 216