JANOWSKI STARLIKE LOG-HARMONIC UNIVALENT FUNCTIONS

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1 Hacettepe Jounal of Mathematics and Statistics Volume , JANOWSKI STARLIKE LOG-HARMONIC UNIVALENT FUNCTIONS Yaşa Polatoğlu and Ehan Deniz Received :0 :008 : Accepted 0 : :008 Abstact Let and be analytic functions in the open unit disc D = {z z < }, with the nomalization h0 = g0 =. The class of log-hamonic mappings of the fom f = z is denoted by S lh. The aim of this pape is to investigate the class of Janowski stalike log-hamonic mappings, a subclass of the log-hamonic mappings. Keywods: Log-hamonic univalent functions, Janowski stalike log-hamonic functions, Subodination pinciple, Distotion theoems. 000 AMS Classification: Pimay 30 C35, 30 C45; Seconday 35Q30.. Intoduction Let HD be the the linea space of all analytic functions defined on the unit disc D. A log-hamonic mapping f is a solution of the non-linea elliptic patial diffeential equation. f z f = wzfz f, whee the second dilatation function wz HD is such that wz < fo all z D. It has been shown that if f is a non-vanishing log-hamonic mapping, then f can be expessed as. f = whee and ae analytic functions in D. On the othe hand, if f vanishes at z = 0 but is not identically zeo, then f admits the epesentation.3 f = z z β, Depatment of Mathematics and Compute Science, İstanbul Kültü Univesity, 3456 İstanbul, Tukey. y.polatoglu@iku.edu.t Depatment of Mathematics Science, Atatük Univesity, Ezuum, Tukey. edeniz36@yahoo.com.t

2 46 Y. Polatoğlu, E. Deniz whee Reβ > /, and ae analytic functions in D, g0 = and h0 0, []. Let us denote by Ω the family of functions φz which ae egula in D and satisfy the conditions φ0 = 0 and φz < fo all z D. Fo abitay fixed eal numbes A and, with < A we use PA, to denote the family of functions.4 pz = + p z + p z + which ae egula in D, and such that pz is in PA, if and only if.5 pz = + Aφz + φz fo some function φz and evey z D. Let S A, denote the class of functions sz = z + c z + which ae analytic in D, such that sz S A, if and only if.6 z s z sz = pz fo evey z D and fo some pz PA, [4]. Let f = z be a univalent log-hamonic mapping. We say that f is a Janowski stalike log-hamonic mapping if Agfe iθ zfz zf z.7 Re = Re > A θ f fo some pz in PA, and all z D [4]. We denote by S lha, the set of all Janowski stalike log-hamonic mappings. Futhe, fo analytic functions S z and S z in D, S z is said to be subodinate to S z if thee exists φz Ω such that S z = S φz fo all z D. We denote this subodination by S z S z. In paticula, if S z is univalent in D, then the subodination S z S z is equivalent to S 0 = S 0 and S D S D [].. Main Results Fo the poof of the main theoem, we need the following lemmas which wee poved by I.S. Jack [3], Kozuo Kuoki and S. Owa [5], espectively... Lemma. Let φz be a non-constant function and analytic in D with φ0 = 0. If φz attains its maximum value on the cicle z = < at a point z 0 D, then we have. z 0φ z 0 = kφz 0 fo some eal k with k... Lemma. Let pz be an element of PA, then. Repz > A 0. The following lemma was poved by H. Silveman and E. M. Silvia [6]..3. Lemma. sz S A, if and only if z s z sz A < A, z D,..4. Theoem. Let f = z be a log-hamonic mapping on D and 0 / hgd. If zh A z z.3 zg z = F z, 0; + z Az = F z, = 0; then f S lha,.

3 Janowski Stalike Log-Hamonic Univalent Functions 47 Poof. We define the function.4 = { + φz A, 0; e Aφz, = 0; whee + φz A has the value at z = 0 We conside the coesponding Riemann banch. Then φz is analytic in D and φ0 = 0. If we take the logaithmic deivative zh A zφ z z.5 zg z, 0; = + φz Azφ z, = 0. Now it easy to ealize that the subodination.3 is equivalent to wz < fo all z D. Indeed, assume the contay. Then thee is z 0 D such that φz 0 =, so by lemma. z 0φ z 0 = kφz 0, k, and fo such z 0 D we have.6 z 0h z 0 hz 0 z0g z 0 gz 0 ka φz 0 = F φz 0 / F D, 0; = + φz 0 kaφz 0 = F φz 0 / F D, = 0; but this contadicts.3; so ou assumptions is wong, i.e, wz < fo evey z D. y using Condition.3 we get.7 + zh z zg z + Aφz = pz, 0; = + φz + Aφz = pz, = 0; and using Lemma., we have.8 Re + zh z zg z On the othe hand.9 = Repz > A. zfz zfz f = z = = + zh z f zg z zfz zf z = Re = Re + zh z f zg z = Re + zh z zg z. Consideing the elations.7,.8,.9 and Lemma.3 togethe, we obtain that f S lha,. The coollay below is a simple consequence of Theoem.3. It is known as the Max-Stohhacke Inequality of f..5. Coollay. A <, 0;.0 log < A, = 0.

4 48 Y. Polatoğlu, E. Deniz Poof. Using.7 we have: A A = φz, 0 = <, 0, log = Aφz, = 0 = log < A, = 0. This completes the poof..6. Theoem. If f S lha, then A A +, 0;. e A ea, = 0. Poof. Using Theoem.3 we have z h z z g z A A., 0, z h z z g z A, = 0. The inequalities. can be witten in the following fom: A + log A log, 0,.3 A log log A, = 0. Then, afte integation we obtain...7. Coollay. If f = S lha,, then b b g z.4 + A h z b b e A g z h z b + a + b., 0, A b + a + b.e A, = 0. Poof. Since f = is the solution of the non-linea elliptic patial diffeential equation fz f = wz.fz, we have f.5 wz = f z.f f f z = g z h z Theefoe we define the function wz w0.6 φz = w0wz = b +. b b b a a a z +. that satisfies the assumptions of Schwaz s Lemma. Using the estimate of Schwaz s Lemma we have φz, which gives.7 wz w0 w0 wz.

5 Janowski Stalike Log-Hamonic Univalent Functions 49 This inequality is equivalent to b.8 wz, and equality holds only fo the function.9 wz = + z + b b z. Fom the inequality.8 we obtain b a wz b a =, b a b + wz + = + b Theefoe we have b b a.0 + wz. + b Consideing.5,.8 and Theoem.6 togethe we obtain.4. Refeences [] Zayid Abdulhadi, Z. and Abu Muhanna, Y. Stalike log-hamonic mappings of ode α, Jounal of Inequalities in Pue and Applied Mathematics 74, 3, 006. [] Goodman, A. W. Univalent Functions, Vol I, II Maine Publ. Comp., Tampa, Floida, 984. [3] Jack, I. S. Functions stalike and convex of ode α, J. London Math. Soc. 3, , 97. [4] Janowski, W. Some extemal poblems fo cetain families of analytic functions, Annales Polinici Mathematici XXVIII, 97 36, 973 elin, 957. [5] Kuoki, K. and Owa, S. Some applications of Janowski functions, Intenational Shot Joint Reseach Wokshop Study On Non-Analytic and Univalent Functions and Applications, Reseach Institute fo Mathematical Science Kyoto Univesity RIMS, May, 008. [6] Silveman, H. and Silvia, E. M. Subclasses of stalike functions subodinate to convex functions, Canad. J. Math. 37, 48 6, 985.

KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS

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