Hacettepe Jounal of Mathematics and Statistics Volume 38 009, 45 49 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. E-mail: y.polatoglu@iku.edu.t Depatment of Mathematics Science, Atatük Univesity, Ezuum, Tukey. E-mail: edeniz36@yahoo.com.t
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,.
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.
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.
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, 469 474, 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.