Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 8 Issue Version.0 Year 208 Tye: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Online ISSN: 2249-4626 & Print ISSN: 0975-5896 Quasi-Hadamard Product of Certain Starlike and Convex P- Valent Functions By R. M. El-Ashwah, A. Y. Lashin & A. E. El-Shirbiny Damietta University Abstract- In this aer, we obtained some results using the quasi-hadamard roduct for two classes of - valent functions related to starlike and convex with resect to symmetric oints. Keywords: starlike function; convex function with resect to symmetric oints; quasi-hadamard roduct. GJSFR-F Classification: MSC 200: 5B34 QuasiHadamardProductofCertainStarlikeandConvexPValentFunctions Strictly as er the comliance and regulations of: 208. R. M. El-Ashwah, A. Y. Lashin & A. E. El-Shirbiny. This is a research/review aer, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unorted License htt://creativecommons.org/licenses/by-nc/3.0/), ermitting all non commercial use, distribution, and reroduction in any medium, rovided the original work is roerly cited.
Notes Quasi-Hadamard Product of Certain Starlike and Convex P-Valent Functions R. M. El-Ashwah α, A. Y. Lashin σ & A. E. El-Shirbiny ρ Abstract- In this aer, we obtained some results using the quasi-hadamard roduct for two classes of - valent functions related to starlike and convex with resect to symmetric oints. Keywords: starlike function; convex function with resect to symmetric oints; quasi-hadamard roduct. I. Introduction let T ) denote of the class of functions of the form: and f z) = a z f r z) = a ;r z g z) = b z a n+ z n+ ; 2 N = f; 2; :::g; a 0; a n+ 0); ) g j z) = b ;j z a n+;r z n+ r 2 N; a ;r > 0; a n+;r 0); 2) b n+ z n+ 2 N = f; 2; :::g; b 0; b n+ 0); 3) b n+;j z n+ j 2 N; b 0; b n+;j 0); 4) which are analytic and -valent in the oen unit disc U = fz : z 2 C; jzj < g: We write T ) = T, the class of analytic functions of the form f z) = a z a n z n ; a 0; a n 0); 5) n=2 which are analytic in the oen unit disc U = fz : z 2 C; jzj < g. Let S be the subclass of functions T consisting of starlike functions in U: It is well known that f 2 S if and only if Global Journal of Science Frontier Research F ) Volume VIII Issue I V ersion I Year 208 99 Author α ρ: Deartment of Mathematics, Faculty of Science, Damietta University, New Damietta 3457, Egyt. e-mails: r_elashwah@yahoo.com, amina.ali66@yahoo.com Author σ: Deartment of Mathematics, Faculty of Science, Mansoura University, Mansoura 3556, Egyt. e-mail: aylashin@mans.edu.eg 208 Global Journals
Global Journal of Science Frontier Research F ) Volume VIII Issue I V ersion I Year 208 00 Quasi-Hadamard Product of Certain Starlike and Convex P-Valent Functions Re zfz) > 0; z 2 U) : f z) and C be the subclass of functions T consisting of convex functions in U: It is well known that f 2 C if and only if Re + zf00 z) > 0; z 2 U): f 0 z) Let S S be the subclass of T consisting of functions of the form 5) satisfying zf 0 z) Re > 0; z 2 U) : f z) f z) These functions are called starlike with resect to symmetric oints and introduced by Sakaguchi [3] see also Robertson [2], Stankiewics [4], Wu [6] and Owa et al. [8]). Khairnar and Rajas see [4], with = 0) introduced the class Ss ; ; ) consisting of functions of the form ) and satisfying the following condition zf 0 z) f z) f z) < zf 0 z) f z) f z) + ; 0 < ; 0 < : Let S c ; ; ) denote the class of functions of the form ) for which zf 0 z) 2 S s ; ; ) : We note that S s ; ; ) = S s ; ) see [5] and [] ) and S c ; ; ) = S c ; ) see [3]). By using the technique of Khairnar and Rajas see [4], with = 0), we get the following theorem. Theorem. Let the function f z) de ned by ) then i) f z) 2 S s ; ; ) if and only if n + + ) + ) ) n+ a n+ [ + ) )] + ) ) a ii) f z) 2 S c ; ; ) if and only if n + n + + ) + ) ) n+ a n+ [ + ) )] + ) ) a iii) f z) 2 Ss;h ; ; ) if and only if h n + + n + ) + ) ) +n a n+ [ + ) )] + ) ) a ; Ref 3. K. Sakaguchi, On certain univalent maing, J. Math. Soc. Jaen., 959), 72-75. where 0 < ; 0 < < ; 0 < number. 2 ) + <, z 2 U and h is a nonnegative real 208 Global Journals
Quasi-Hadamard Product of Certain Starlike and Convex P-Valent Functions We note that for a nonnegative real number h the class Ss;h ; ; ) is nonemty as the function of the form f z) = a z h h n+ [ + ) )] + ) + ) + ) ) n+ i a n+ z n+ ; 6) +n Ref 2. H. E. Darwish, A. Y. Lashin and A. N. Alnayyef, Quasi Hadmeard roduct of certain starlike and convex functions, Global Journal of Science Frontier Research, 5 0) 205), no., -7. where a > 0; n+ 0 and n+ satisfy the inequality 6). It is evident that S s; ; ; ) = S c ; ; ) and for h = 0; S s;0 ; ; ) is identical to S s ; ; ) : Further more S s;h ; ; ) S s;m ; ; ) for h > m; the containment being roer. Hence for any ositive integer h, the inclusion relation S s;h; ; )S s;h ; ; )::: S s;m; ; ):::S s;2 ; ; ) S c ; ; )S s ; ; ) : The quasi-hadamard roduct of two or more functions has recently de ned by Darwich et al.[2], Kumar[5, 6, 7], Owa [9, 0, ], and others. Accordingly, the quasi-hadmard roduct of two functions f z) and g z) f z) g z) = a b z a n+ b n+ z n+ : In this aer, we obtained some results concerning the quasi-hadamard roduct for two classes of valent functions related to starlike and convex with resect to symmetric oints. Theorem 2 A functions f r z) de ned by 2) in the Sc ; ; ) for each r = ; 2; :::; u: then we get quasi-hadamard rouduct Proof. f z) f 2 z) ::: f u z) 2 S s;2u )+ ; ; ) : To rove the theorem, we need to show that n + 2u Since f r z) 2 S c ; ; ), we have II. Main Results )+ + n + ) + ) ) n+ [ + ) )] + ) ) a ;r : a n+;r n + + n + ) + ) ) n+ a n+;r [[+ ) )]+ ) ]a ;r 7) Global Journal of Science Frontier Research F ) Volume VIII Issue I V ersion I Year 208 0 208 Global Journals
Quasi-Hadamard Product of Certain Starlike and Convex P-Valent Functions for each r = ; :::; u: Therefore n + + n + ) + ) ) n+ a n+;r [[+ ) )]+ ) ]a ;r or 8 < a n+;r : h n+ 9 [ + ) )] + ) = + ) + ) ) n+ i ; a ;r +n Notes Global Journal of Science Frontier Research F ) Volume VIII Issue I V ersion I Year 208 02 for each r = ; 2; :::; u. The right hand exression of this last inequality is not greater than 2 a;r ; hence n+ By 8) for each r = ; 2; :::; u n + " n + = " u 2u 2u and 7) for r = u; we get )+ + n + ) + ) ) n+ a n+;r )+ + n + ) + ) ) n+ n + 2u ) u Y n + + n a ;r + ) + ) ) n+ a n+;u [ + ) )] + ) ) a ;r hence f f 2 ::: f u 2 S2u )+ ; ; ). This comletes the roof of Theorem 2. 8) Y a ;r )a n+;u Theorem 3 A functions f r z) de ned by 2)in the class Ss ; ; ) for each r = ; 2; :::; u: Then, we get the quasi-hadamard roduct Proof. Using f r z) 2 Ss ; ; ) ; we have f z) f 2 z) ::: f u z) 2 S s;u ) ; ; ) : + n + ) + ) ) n+ a n+;r [ + ) )] + ) a ;r 9) for each r = ; 2; :::; u; therefore, 0 [ + ) )] + ) a n+;r @ 2 n + a n+;r a ;r r = ; :::; u: +n + ) + ) ) n+ A a ;r 208 Global Journals
Quasi-Hadamard Product of Certain Starlike and Convex P-Valent Functions and hence n + a n+;r a ;r; r = ; 2; :::; u: 0) Notes By 0) for r = ; 2; :::; u n + n + " u = u u ) and 9) for r = u; we get ) + n + ) + ) ) n+ ) a n+;r ) + n + ) + ) ) n+ " n + Y + n a ;r + ) + ) ) n+ a n+;u [ + ) ] + ) ) a ;r u ) u Y a ;r )a n+;u Hence f z) f 2 z) ::: f u z) 2 S s;u ; ; ). This comletes the roof of Theorem 3. Theorem 4 A functions f r z) de ned by 2) in the class Sc ; ; ) for each r = ; 2; :::; u and the functions g j z) defined by.4) in the class Ss ; ; ) for j = ; 2 ; :::; q: Then, we get the quasi- Hadamard roduct ; f z) f 2 z) ::: f u z) g z) g 2 z) ::: g q z) 2 S s;2u+q ; ; ): Proof. We denote the quasi-hadamard roduc f z)f 2 z):::f u z)g z)g 2 z):::g q z) by function h z) ; for the sake of the convenience. Clearly " " h z) = a ;r : b ;j z a n+;r : b n+;j z n+ : To rove the theorem, we need to show that n 2u+q + + n + ) + ) ) n+ " a n+;r : " [ + ) )] + ) ) a ;r : b ;j ) b n+;j Since f r z) 2 Sc ; ; ) ; the inequalities 7) and 8) hold for every r = ; 2; :::; u: Further, since g j z) 2 Ss ; ; ) ; the inequality 0) holds for each j = ; 2; :::; q. By 8) for r = ; 2; :::; u; 0) for j = ; 2; :::; q and 9) for j = q, we get n " 2u+q + + n + ) + ) ) n+ " ) a n+;r : b n+;j ) Global Journal of Science Frontier Research F ) Volume VIII Issue I V ersion I Year 208 03 208 Global Journals
Global Journal of Science Frontier Research F ) Volume VIII Issue I V ersion I Year 208 04 = n + n + 2u n + 2u+q a ;r : 2u+q + n! b n+;j ) + n " q Y 2u q ) n + n + a ;r : b ;j : + ) + ) ) n+ ) + ) + ) ) n+ ) b n+;q " q Y + n a ;r : b ;j + ) + ) ) n+ b n+;q [ + ) )] + ) ) " a ;r b ;j : Hence, h z) 2 S s;2u+q ; ; ): This comletes the roof of Theorem 4. Remark 5 al. [2]. Quasi-Hadamard Product of Certain Starlike and Convex P-Valent Functions Putting = in the above results, we obtain the results obtained by Darwish et References Références Referencias. M. K. Aouf, R. M. El-Ashwah and S. M. El-Deeb, Certain classes of univalent functions with negative coefficients and n-starlike with resect to certain oints, Matematiqki Vesnik, 62 200),no. 3, 25.226. 2. H. E. Darwish, A. Y. Lashin and A. N. Alnayyef, Quasi Hadmeard roduct of certain starlike and convex functions, Global Journal of Science Frontier Research, 5 0) 205), no., -7. 3. W. S. Jiuon and A. Janteng, Classes with negative coefficients and convex with resect to other oints, Int. J. Contem. Math. Sciences, 3 2008), no., 5-58. 4. S. M. Khairnar and S. M. Rajas, Proerties of a class of multivalent functions starlike with resect to symmetric and conjugate oints, General Mathematics, 8 200), no. 2,5-29. 5. V. Kumar, Hadamard roduct of certain starlike functions, J. Math. Anal. Al., 0 985), 425-428. 6. V. Kumar, Hadamard roduct of certain starlike functions II, J. Math. Anal. Al., 3 986), 230-234. 7. V. Kumar, Quasi Hadamard roduct of certain univalent function, J Math. Anal. Al. 26 987), 70-77. 8. S. Owa, Z. Wu and F. Ren, A note on certain subclass of Sakaguchin functions, Bull. Soc. Roy. Liege, 57 988), 43-50. Ref 2. H. E. Darwish, A. Y. Lashin and A. N. Alnayyef, Quasi Hadmeard roduct of certain starlike and convex functions, Global Journal of Science Frontier Research, 5 0) 205), no., -7. 208 Global Journals
Quasi-Hadamard Product of Certain Starlike and Convex P-Valent Functions Notes 9. S. Owa, On the classes of univalent functions with negative coefficients, Math. Jaon., 27 4) 982), 409-46. 0. S. Owa, On the starlike functions of order αα and tye ββ, Math. Jaon., 27 6) 982), 723-735.. S. Owa, On the Hadamard roduct of univalent functions, Tamkang J. Math., 4 983), 5-2. 2. M. S. Robertson, Alications of the subordination rincile to univalent functions, Pacific J. Math., 96), 35-324. 3. K. Sakaguchi, On certain univalent maing, J. Math. Soc. Jaen., 959), 72-75. 4. J. Stankiewicz, Some remarks on functions starlike with resect to symmetric oints, Ann. Univ. Marie Curie Sklodowska, 9 965), 53-59. 5. T.V. Sudharsan, P. Balasubrahmananyam, K.G. Subramanian, On functions starlike with resect to symmetric and conjugate oints, Taiwanese J. Math. 2 998), 57.68. 6. Z. Wu, On class of Sakaguchi functions and Hadamard roducts, Sci. Sinica Ser. A, 30987), 28-35. 05 Global Journal of Science Frontier Research F ) Volume VIII Issue I V ersion I Year 208 208 Global Journals
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