Inclusion and Argument Properties for Certain Subclasses of Analytic Functions Defined by Using on Extended Multiplier Transformations

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1 Advances in Pure Matheatics, 0,, doi:0.436/a Pubished Onine Juy 0 (htt:// Incusion Arguent Proerties for Certain Subcasses of Anaytic Functions Defined by Using on Extended Mutiier Transforations Abstract Oh Sang Kwon Deartent of Matheatics, Kyungsung University, Busan, Korea E-ai: oskwon@ks.ac.kr Received March 8, 0 revised Ari 7, 0 acceted May 5, 0 Making use of a utiier transforation, which is defined by eans of the Hadaard roduct (or convoution), we introduce soe new subcasses of anaytic functions investigate their incusion reationshis arguent roerties. Keywords: Subordination, Starike Functions, Convex Functions, Cosed-to-Convex Functions, Mutiier Transforation, Mutivaent Functions, Arguent Princie. Introduction Let A denote the cass of functions f noraized by k k k f z z a z ( : {,,3, }) (.) which are anaytic -vaent in the oen unit disk U z: z z < If f g are anaytic in U, we say that f is subordinate to g, write z 0 0 z < for zu. * We denote by S C f g or f z g z ( zu ) if there exists a Schwarz function, anaytic in U in zu, such that f z g z the subcasses of A consisting of a anaytic functions which are, resectivey, -vaent starike of order (0 < ) in U -vaent convex of order (0 < ) in U. Let M be the cass of anaytic functions 0, which are convex univaent in U satisfy the foowing inequaity: z Re > 0 ( zu ) Making use of the aforeentioned rincie of subordination between anaytic functions, we define each of the foowing subcasses of A : S * z zf : Re f : f A z f z (.) (0 < z U M) K z zf : Re f : f A z f z (0 < z U M) (.3) For : {0,,, } 0, we define the utiier J,, of functions f A by transforation z * zf C,, : Re f : f A gs st.. g z (0, < z U, M) z (.4) Coyright 0 SciRes.

2 94 O. S. KWON k k J,, f z z ak z k ( > 0 0 zu) Put k,, z z z k ( > 0 0 zu) k (.5) (.6) The oerators,,,,, are the utiier transforations introduced studied earier by Sarangi Uraegaddi [6] Uraegaddi Soanatha ([] []), resectivey. Corresending to the function defined by (.6), we introduce a function,, z,,, z given by the Hadaard roduct (or convoution):, z,, z*,, z ( > ) z Then, anaogous to J,, new utiier transforation as foows: We note that I,, : A A,,, * I f z,, z f z, we have define a (.7),,,, z I f z f z I f z zf 0 It is easiy verifed fro the above definition of the oerator I,,, that,, z I f z I,, f z I,, f z,, z I f z I,, f z I,, f z (.8) (.9) The definition (.6) of the utiier transforation,, is otivated essentiay by the Choi-Saigo- Srivastava oerator [3] for anaytic functions, which incudes a sier integra oerator studied earier by Noor [7] others (cf. [4-6]). Next, by using the oerator I,, defined by (.7), we introduce the foowing subcasses of anaytic functions: K, S,, * f : f A I,, f z S,,, ( M,, > 0 0 <) (.0) f : f A I,, f z K ( M,, > 0 0 <), C,,,, f : f A I,, f z C,, (.) (, M,, > 0 0, <) (.) We aso note that,, zfzs f z K (.3),,,, In articuar, we set, Az, S,,,,, ( < < ) S A B B A Bz (.4), Az, K,, K,, A, B ( < B < A) Bz (.5) In the resent aer, we investigate soe incusion reationshis arguent roerties associated such utivaent functions in the cass A as those be, onging to the subcasses S,,,, K,,, C,,,, defined by (.0), (.) (.), resectivey.. Incusion Proerties Lea.: Let be convex univaent in U 0 Re z > 0 (, ). If is n U 0, then anaytic i z z z ( ) z z z U iies that z z Theore.: Let (z U ). M in Re z> ax, zu Coyright 0 SciRes.

3 O. S. KWON 95 then,, S S S,.,,,,,, Proof. First of a, we show that, S S,, f S set. Let,,,, z z I f z I f z where the function 0. Aying (.), we obtain,,,, z,,,, z,, is anaytic in U (.) I f z z (.) I f By ogarithicay differentiating both sides of (.) utiying the reseuting equation by z, we have z I,, f z I,, f z (.3) zz z z ( zu) Since Re z > 0, by aying Le- a. to (.3), it foows that z z in U, that, is, that f z S,,. To rove the second art of Theore., et, f ut z S,, q z z I f z I f z,,,, where the function q z is anaytic in U q 0. In recisey the sae anner, we can find the resut that qz z in U, that is, that, f z S,, under the hyothesis Re z > 0 Theore.3: Let z M in Re z>ax, zu then,,, K,, K,, K,,. Proof. Aying (.) Theore., we observe that,,,,,,,, zf z S f z K f z K zf z S,,,,,,,,,, f z K zf z S,, zf z S f z K,,,, which evidenty rove Theore.3. By setting Az z ( < B < A z U) Bz in Theores..3, we deduce the foowing coroary. Coroary.4: Suose that A >ax, B Then, for the function casses defined by (.) (.3), then,,, S,, S,, A B S,, A B,, A, B,,,,,,,,,,,, K AB K AB K AB Theore.5: Let, M in Re z>ax, zu C C, C,,,,,,,,,,,,,, Proof. We begin by roving that,,,, C C,,, which is the first,,,, incusion reationshi asserted by Theore.5., Let f z C,,,,. Then there exists a k z S * such that function z I,, f z z ( z U) kz Choose the function g z such that * z S I,, g z k Coyright 0 SciRes.

4 96 O. S. KWON, Then, g z S,, z I f z I g z,,,,, S,, z ( zu) (.4) Now et z where the function 0. z I f z I g z Using (.9), we find that,,,, z is anaytic in U (.5),,,, z I,, f z z z I f z I f z I,, gz zi,, gz I,, gz zi f z z I f z,,,,,,,, z I g z I g z z I,, f z z I,, f z I,, gz I,, gz zi,, gz I,, gz, Since gz S q z,,, then we set z I,, f z I,, gz (.6) w here q z z in U the assution that M. By (.5),,,,, z I f z I g z z Differentiating both side of (.7) resect to z u tiying by z, we obtain Hence,, z I f z,, I f z qz zz z,, I,, gz zz qz z z I f z (.7) Couting the above equations, we can obtain (.8) z I f z I g z,,,, z zz zz qz q z z q z z Coyright 0 SciRes.

5 O. S. KWON 97 Since Re z > 0, aying Lea, we can show. wz z, that z z in U, so that f z C 3. Arguent Proerties,,,,. Lea 3.: Let be convex univaent in U be anay tic in U Re z 0. If z is 0 0, then anaytic in U zzz z ( z zu ) iies that z z (z U ). Lea 3.: Let be anaytic in U 0 z0 for a zu. If there exist o oints z, z U such that tw arg <arg (3.) ) f o r a z z <argz z for soe (, >0 ( z < z z ). z z z z i i z z (3.) b where b i. b 4 Theore 3.3: Let f A. 0<,. 0< <. If for soe z I,, f z <arg < I,, gz, g S,,, A, B, then <arg z I,, f z < I,, g z where, are the soutions for the foowing equatio ns: ( ) b c os t A b bsin t B ( ) b cos t A b bsin t B b is given by (3.), t t AB cos AB B (3.3) z I,, f z Proof. Let z I,, gz. Then z is anaytic in U 0. By using (.9), we obtain z I,, gz,,,, I f z I f z (3.4) Differentiating both sides of the above equation utiying the resuting equation by z, we find that z I,, gz ( z) I,, gz I,, f z I,, f z, Since g z S,,, A, B, foows that g z S A Next we et,,,, B. q z, by Coroary.4, it z I,, gz I,, gz. Coyright 0 SciRes.

6 98 O. S. KWON Then, using (.9), we have,,,, I g z qz (3.5) I g z Fro (3.4) (3.5), we obtain zz qz z I,, f z I,, gz z Furtherore, by using a known resut, we have AB < A q z B (3.6) B B Thus, fro (3.6), we obtain i qz ex where, in ters of t given by (3.3). A A < < B B t < < t We note that is anaytic in 0. Let h z be the function which as U onto the angu ar doain U : < arg < h0 Aying Lea 3. for this function z qz h we see that Re z > 0 ( zu ), hence z 0 (z U ). By using Le a 3., if there exist two oints z, z U such that the condition (3.) is satisfied, then we obtain (3.) under the constraint (3.). And we obtain arg z z z qz cos i i cos arg ex bcos t A b bcos t B bcos t arg zz π z q z A b b cos t which woud obviousy contradict the assertion of Theore 3.3. We thus coete the roof of Theore 3.3. If we et in Theore 3.5, we easiy obtain the foowing consequence. Coroary 3.4: Let f A. 0<. 0< <. If z I,, f z arg < I,, gz B for soe, g S,,, A, B, then z I f z arg I g z <,,,, where is the soutions for the foowing equation: Coyright 0 SciRes.

7 O. S. KWON 99 bcos t A b bcos t B b is given by (3.), z I,, f z <arg < I,,gz (3.7) Theore 3.5: Let f A. 0<,. 0< <. If z I,, f z <arg < I,, gz, for soe g S A,,,, B, then z I,, f z <arg I,, gz < where, are the soutions for the foowing equations: bcos t A b bcos t B bcos t A b bcos t B b is given by (3.), zi,, f z t t arg < I,, gz ( AB) cos, AB B for soe g S,,, A, B, then (3.8) zi,, f z arg < If we et in Theore 3.5, we easiy obtain I,, gz the foowing consequence. Coroary 3.6: Let f A. 0<. 0< <. If where is the soutions for the foowing equation: bcos t A b bcos t B Coyright 0 SciRes.

8 00 O. S. KWON b is given by (.7), t t cos 4. Acknow edgeents AB AB B (3.9) The research was suorted by Kyungsung University Research Grants in References [] B. A. Uraegaddi C. Soanatha, Certain Differentia Oerators for Meroorhic Functions, Houston Journa of Matheatics, Vo. 7, 99, [] B. A. Uraegaddi C. Soanatha, New Critetia for Meroorhic Starike Functions, Buetin of the Austraian Matheatica Society, Vo. 43, No., 99, doi:0.07/s [3] J. H. Choi, M. Saigo H. M. Srivastava, Soe Incusion Proerties of a Certain Faiy of Integra Oera- tors, Journa of Matheatica Anaysis Aications, Vo. 76, No., 00, doi:0.06/s00-47x(0) [4] J.-L. Liu K. I. Noor, Soe Proerties of Noor Integra Oerator, Journa of Natura Geoetry, Vo., 00, [5] J.-L. Liu, The Noor Integra Strongy Starike Functions, Journa of Matheatica Anaysis Aications, Vo. 6, No., 00, doi:0.006/jaa [6] K. I. Noor M. A. Noor, On Integra Oerators, Journa of Matheatica Anaysis Aications, Vo. 38, No., 999, doi:0.006/jaa [7] K. I. Noor, On New Casses of Integra Oerators, Journa of Natura Geoetry, Vo. 6, 999, [8] K. S. Padanabhan R. Parvatha, On Anaytic Functions Differentia Subordination, Buetin Mathéatique de a Société des Sciences, Mathéatiques de Rouanie, Vo. 3, 987, [9] M. Nunokawa, S. Owa, H. Saitoh, N. E. Cho N. Ta-kahashi, Soe Proerties of Anaytic Functions at Extrea Points for Arguents, rerint, 003. [0] P. Eenigenburg, S. S. Mier, P. T. Mocanu M. O. Reade, On a Briot-Bouquet Differentia Subordination, Genera Inequaities, Vo. 3, 983, [] R. J. Libera M. S. Robertso n, Meroorhic Coseto-Convex Functions, Michigan Matheatica Journa, Vo. 8, No., 96, doi:0.307/j/ [] S. K. Bajai, A Note on a Cass of Meroorhic Univaent Functions, Revue Rouaine de Mathéatiques Pures et Aiquées, Vo., 997, [3] S. M. Sarangi S. B. Uraegaddi, Certain Differentia Oerators for Meroorhic Functions, Buetin of the Cacutta Matheatica Society, Vo. 88, 996, [4] S. S. Mier P. T. Mocanu, Differentia Subordinations Univaent Functions, Michigan Matheatica Journa, Vo. 8, No., 98, Coyright 0 SciRes.