QuasiHadamardProductofCertainStarlikeandConvexFunctions

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1 Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 15 Issue 10 Version 1.0 Year 2015 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA Online ISSN: & Print ISSN: Quasi-Hadamard Product of Certain Starlike and Convex Functions By H. E. Darwish, A. Y. Lashin & A. N. Alnayyef Mansoura University Mansoura, Egypt Abstract- In this paper, we establish certain results concerning the quasi-hadmard product for two classes related to starlike and convex univalent functions with respect to symmetric points. Keywords: starlike and convex functions with respect to symmetric points, quasi-hadamard product. GJSFR-F Classification : MSC 2010: 30C45 QuasiHadamardProductofCertainStarlikeandConvexFunctions Strictly as per the compliance and regulations of : H. E. Darwish, A. Y. Lashin & A. N. Alnayyef. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Notes Quasi-Hadamard Product of Certain Starlike and Convex Functions H. E. Darwish α, A. Y. Lashin σ & A. N. Alnayyef ρ Abstract- In this paper, we establish certain results concerning the quasi-hadmard product for two classes related to starlike and convex univalent functions with respect to symmetric points. Keywords: starlike and convex functions with respect to symmetric points, quasi-hadamard product. I. Introduction X V Year Global Journal of Science Frontier Research Volume XV Issue ersion I Throughout this paper, let S denote of the functions of the form : F (z = a 1 z a k z k (a 1 > 0, a k 0, (1.1 F r (z = a 1,r z a k,r z k (r N, a 1,r > 0, a k,r 0, (1.2 F and ğ(z = b 1 z b k z k (b 1 > 0, b k 0 ğ j (z = b 1,j z b k,j z k (j N, b 1,j > 0, b k,j 0 (1.3 (1.4 which are analytic in the unit disc U = {z : z < 1}. Let S be the subclass of functions S consisting of starlike functions in U. It is well known that F S if and only if { } zf (z Re > 0, (z U, F (z (1.5 Author α σ ρ: Department of Mathematics Faculty of Science, Mansoura University Mansoura, 35516, Egypt. s: darwish333@yahoo.com, aylashin@mans.edu.eg, ahmed.nabeel36@yahoo.com

3 and C be the subclass of functions S consisting of convex functions in U. It is well known that F C if and only if } zf (z Re {1 + > 0, (z U. F (z (1.6 Global Journal of Science Frontier Research F Volume XV Issue X V ersion I Year Let Ss be the subclass of S consisting of functions of the form (1.1 satisfying { } zf (z Re > 0, (z U. (1.7 F (z F ( z These functions are called starlike with respect to symmetric points and were introduced by Sakaguchi 10] ( see also Robertson 9], Stankiewics 12] Wu 14] and Owa et al. 5]. In 13], Sudharsan et al. introduced the class Ss (α, β consisting of functions of the form (1.1 and satisfying the following condition (see also 11] zf (z F (z F ( z 1 < β zf (z α F (z F ( z + 1 for some 0 α 1, 0 < β 1 and z U. Let S c (α, β denote the class of function F (z of the from (1.1 for which zf S s (α, β. By using the same technique of Sudharsan et al. 13] and Aouf et al. 1], we get the following theorem. Theorem 1. Let the function F (z defined by (1.1. Then (i F (z S s (α, β if and only if (1 + αβ k + (β 1 1 ( 1 k] ] a k,r β (2 + α 1] a 1,r where 0 α 1, 0 < β 1, 0 2(1 β 1+αβ (ii F (z S c (α, β if and only if < 1 and z U. k (1 + αβ k + (β 1 1 ( 1 k] ] a k,r β (2 + α 1] a 1,r where 0 α 1, 0 < β 1, 0 2(1 β 1+αβ (iii F (z Ss,h (α, β if and only if < 1 and z U. k {(1 h + αβ k + (β 1 1 ( 1 k]} a k,r β (2 + α 1] a 1,r, (1.8 (1.9 (1.10 (1.11 Ref 5. S. Owa, Z. Wu and F. Ren, A note on certain subclass of Sakaguchin functions, Bull. Soc. Roy. Liege 57 (1988,

4 Ref 2. V. Kumar, Hadamard product of certain starlike functions, J. Math. Anal. Appl. 110 (1985, αβ where 0 α 1, 0 < β 1, 0 < 1 and z U. Where h is an nonnegative real number. We note that for every nonnegative real number h, the class Ss,h (α, β is nonempty as the functions of the from F (z = a 1 z β (2 + α 1 k (1 h + αβ k + (β 1 1 ( 1 k]]a 1λ k z k, where a 1 > 0, λ k 0,and λ k 0, satisfy the inequality (1.12. It is evident that S1(α, β Sc (α, β and, for c = 0, Sc (α, β is identical to S0(α, β. Further, Sc (α, β Sk (α, β if c > k, the containment being proper. Hence, for any positive integerc, the inclusion relation S c (α, β S c 1(α, β... S 2(α, β S c (α, β S s (α, β. The quasi-hadamard product of two or more functions has recently been defined and used by Owa 6, 7, 8], Kumar 2, 3, 4] and others. Accordingly, the quasi-hadamard product of two functions F (z and ğ(z is given by F ğ(z = a 1 b 1 z a k b k z k. Theorem 2. A functions F i (z defined by (1.2 in the class Sc (α, β for each r = 1, 2,..., u.then we get the quasi-hadamard product F 1 F 2... Fu(z S2(u 1+1 (α, β. Proof. To prove the theorem, we need to show that k {(1 2(u αβ k + (β 1 1 ( 1 k]} Π m r=1a k,r β (2 + α 1] a 1,r. Since F r (z S c (α, β,we have { k (1 + αβ k + (β 1 1 ( 1 k]} a k,r β (2 + α 1] a 1,r, for each r = 1, 2,..., u.therefore, or II. The Main Theorems { k (1 + αβ k + (β 1 1 ( 1 k] } a k,r β (2 + α 1] a 1,r β (2 + α 1] a k,r ( k (1 + αβ k + (β 1 1 ( 1 k] a 1,r (1.12 (2.1 (2.2 X V Year Global Journal of Science Frontier Research Volume XV Issue ersion I F

5 Global Journal of Science Frontier Research F Volume XV Issue X V ersion I Year for each r = 1, 2,..., u. The right-hand expression of this last inequality is not greater then k 2 a 1,r.. Hence for each r = 1, 2,..., u. a k,r k 2 a 1,r. By (2.3 for each r = 1, 2,..., u 1, and (2.2 for r = u, we get k {(1 2(u αβ k + (β 1 1 ( 1 k]} Π u r=1a k,r {k 2(u 1+1 (1 + αβ k + (β 1 1 ( 1 k] ] } k 2(u 1 Π u 1 r=1a 1,r a k,u = ] { Π u 1 r=1a 1,r k (1 + αβ k + (β 1 1 ( 1 k] ]} a k,u β (2 + α 1] Π u r=1a 1,r ]. Hence F 1 F 2... F u (z S2(u 1+1 (α, β.this completes the proof of Theorem 2. Theorem 3. A functions F r (z defined by (1.2 in the class Ss (α, β for each r = 1, 2,..., u. Then we get the quasi-hadamard product F 1 F 2... F u (z S(u 1 (α, β. Proof. Using F r (z S s (α, β,we have (1 + αβ k + (β 1 1 ( 1 k] ] a k,r β (2 + α 1] a 1,r for each r = 1, 2,..., u. Therefore, β (2 + α 1] a k,r (1 + αβ k + (β 1 1 ( 1 k] a 1,r and hence for every r = 1, 2,..., u. a k,r k 1 a 1,r By (2.5 for r = 1, 2,..., u 1,and (2.4 for r = u, we get {k (u 1 (1 + αβ k + (β 1 1 ( 1 k] Π ur=1a ]} k,r (2.3 (2.4 (2.5 Notes

6 {k (u 1 (1 + αβ k + (β 1 1 ( 1 k] ] ] } k (u 1 Π u 1 r=1a 1,i a k,u = ] { Π u 1 r=1a 1,r (1 + αβ k + (β 1 1 ( 1 k]] } a k,u Notes β (2 + α 1] Π u r=1a 1,r ] Hence F 1 F 2... F u (z S(u 1 (α, β. This completes the proof of Theorem 3. Theorem 4. A functions Fr (z defined by (1.2 in the class Sc (α, β for each r = 1, 2,..., u; and the functions ğ j (zin the class Ss (α, β for every j = 1, 2,..., q. Then we get the Hadamard product F 1 F 2... F u ğ 1 ğ 2... ğ q (z S2u+q 1(α, β. Proof. We denote the quasi-hadamard product F 1 F 2... F u ğ 1 ğ 2... ğ q (z by the function h (z, for the sake of the convenience. Clearly, h(z = Π u r=1a 1,i.Π q j=1 b ] 1,j z Π u r=1 a k,r.π q j=1 b ] k,j z k. To prove the theorem, we need to show that { ( (k 2u+q 1 (1 + αβ k + (β 1 1 ( 1 k] Π u r=1 a k,r.π q j=1 b ] } k,j X V Year Global Journal of Science Frontier Research Volume XV Issue ersion I F β (2 + α 1] ( Π u r=1a 1,r.Π q j=1 b 1,j. (2.6 Since F r (z S c (α, β,the inequalities (2.2 and (2.3 hold for every r = 1, 2,..., u. Further, since ğ j (z S s (α, β,we have (1 + αβ k + (β 1 1 ( 1 k] ] b k,r β (2 + α 1] b 1,r for each j = 1, 2,..., q. Whence we obtain for each j = 1, 2,..., q. b k,j k 1 b 1,j By (2.3 for r = 1, 2,..., u, (2.8 for j = 1, 2,..., q 1, and (2.7 for j = q, we get { k 2u+q 1 (1 + αβ k + (β 1] 1 ( 1 k] Π u r=1 a k,r.π q j=1 b ] } k,j (2.7 (2.8

7 {k 2u+q 1 (1 + αβ k + (β 1 (1 ( 1 k]} k ( u(2 Π u r=1a 1,r.Π q j=1 b k,j {k 2u (q 1 (1 + αβ k + (β 1 (1 ( 1 k]} Notes Global Journal of Science Frontier Research F Volume XV Issue X V ersion I Year k 2u.k (q 1 Π u r=1a 1,r.Π q 1 j=1 b 1,j] bk,q = Π u r=1a 1,r.Π q 1 j=1 b 1,j ] β (2 + α 1] Π u r=1a 1,r.Π q 1 j=1 b 1,j]. ( (1 + αβ k + (β 1 1 ( 1 k] b k,q Hence h(z S 2u+q 1(α, β. This completes the proof of Theorem 4. References Références Referencias 1. M. K. Aouf, R. M. El-Ashwah and S. M. El-Deeb, Certain classes of univalent functions with negative coeff cients and n-starlike with respect to certain points; Math. vesnik. 62 (3 (2010, V. Kumar, Hadamard product of certain starlike functions, J. Math. Anal. Appl. 110 (1985, V. Kumar, Hadamard product of certain starlike functions II, J. Math. Anal. Appl. 113 ( V. Kumar, Quasi-Hadamard product of certain univalent function, J. Math. Anal. Appl. 126 ( S. Owa, Z. Wu and F. Ren, A note on certain subclass of Sakaguchin functions, Bull. Soc. Roy. Liege 57 (1988, S. Owa, On the classes of univalent functions with negative coeffcients, Math. Japon. 27(4 ( S. Owa, On the starlike functions of order α and type β, Math. Japon. 27 (6 ( S. Owa, On the Hadamard product of univalent functions, Tamkang J. Math. 14 ( M. S. Robertson, Applications of the subordination principle to univalent functions, Pacific J. Math. 11(1961, K. Sakaguchi, On certain univalent mapping, J. Math. Soc. Japen. 11 (1959, J. Sokol, Some remarks on the class of functtions starlike with respect to symmetric points, Folia Scient. Univ. Tech. Resoviensis 73 (1990, J. Stankiewicz, Some remarks on functions starlike with respect to symmetric points, Ann. Univ. Marie Curie Sklodowska 19 (1965,

8 13. T. V. Sudharsan, P. Balasubrahmmanayam and K. G. Subramanian, On functions starlike with respect to symmetric and conjugate points, Taiwanese J. Math. 2 (1998, Z. Wu, On class of Sakaguchi functions and Hadamard products, Sci. Sinica Ser. A 30 (1987, Notes F X V Year Global Journal of Science Frontier Research Volume XV Issue ersion I

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