Int. Journal of Math. Analysis, Vol. 7, 203, no., 9-29 Notes on Starlike log-harmonic Functions of Order α Melike Aydoğan Department of Mathematics Işık University, Meşrutiyet Koyu Şile İstanbul, Turkey melike.aydogan@isikun.edu.tr Emel Yavuz Duman Department of Mathematics Computer Science İstanbul Kültür University İstanbul, Turkey e.yavuz@iku.edu.tr Shigeyoshi Owa Department of Mathematics Kinki University Higashi-Osaka, Osaka 577-8502, Japan shige2@ican.zaq.ne.jp Abstract For log-harmonic functions fz =zhzgz in the open unit disk U, two subclasses HLH α G LH α ofs LH α consisting of all starlike log-harmonic functions of order α 0 α < are considered. The object of the present paper is to discuss some coefficient inequalities for hz gz.
20 M. Aydoğan, E. Yavuz Duman Sh. Owa Mathematics Subject Classification: Primary 30C55, Secondary 30C45 Keywords: Analytic, log-harmonic, starlike, coefficient inequality. Introduction Let H be the class of functions which are analytic in the open unit disc U = {z C : z < }. A log-harmonic function fz is a solution of the non-linear elliptic partial differential equation. f z f = wzf z f, where wz H satisfies wz < z U is said to be the second dilatation,.2 f z = 2 Let a function fz given by f x i f, f z = f y 2 x + i f. y.3 fz =zhzgz with 0 / hgu be log-harmonic function in U, where hz H gz H. Then fz is said to be starlike log-harmonic function of order α if it satisfies.4 arg fre iθ θ zfz zf z =Re >α z U f for some real α 0 α<. We denote by SLH α all starlike log-harmonic functions fz of order α in U. The class SLH α was studied by Abdulhadi Muhanna [4], Polatoğlu Deniz [6]. Furthermore, the classes of univalent log-harmonic functions have been studied by Abdulhadi [], [2], Abdulhadi Hengartner [3].
Notes on starlike log-harmonic functions 2 2 Coefficient Inequalities for hz In order to consider our problem, we have to introduce the following subclass HLH α ofs LH α. A function fz =zhzgz S LH α is said to be in a class HLH α if it satisfies 2. hz =h0 + with a n = a n e inθ+π θ R. a n z n h0 > 0 Now we derive Theorem 2.. If f = zhzgz HLH α with 2.2 zg β < min z U Re z < 0, gz then 2.3 n + α β a n α β h0. Proof. Note that f = zhzgz HLH α S LH α satisfies θ arg zfz zf z freiθ = Re f =Re + zh z hz zg z >α z U. gz This gives us that zh z 2.4 Re =Re na nz n hz h0 + a nz n =Re n a n e inθ z n h0 a n e inθ z n > Re α + zg z gz
22 M. Aydoğan, E. Yavuz Duman Sh. Owa >α+ β for all z U. Let us consider a point z such that z = z e iθ U. Then 2.4 becomes that zh z 2.5 Re = n a n z n hz h0 a n z >α+ β n z U. Letting z, we obtain that n a n α + β h0 a n, that is, that n + α β a n α β h0. Example 2.2. Let us consider a function fz =zhzgz HLH α with α β h0e inθ hz =h0 + nn + n + α β zn Then gz = 2β z 0 > min z U Re zg z gz β < 0. >β n + α β a n = α β h0 nn + = α β h0. Theorem 2. gives us the following corollary. Corollary 2.3. If fz =zhzgz HLH α with 2.2 then a n α β n + α β h0 n =, 2, 3,.
Notes on starlike log-harmonic functions 23 Next, we show Theorem 2.4. If fz =zhzgz HLH α with 2.2 then 2.6 α β z h0 hz + α β z h0 2 α β 2 α β 2.7 a α βh0 2 α β a z h z a + α βh0 2 α β a z for z U. The equality in 2.6 holds for fz =zhzgz with hz =h0 + α β 2 α β h0e iθ z. Proof. We note that the inequality 2.3 gives us that a n α β 2 α β h0 n a n α β h0 2 α β a. n=2 Thus, we have that hz h0 + z a n + α β z h0 2 α β hz h0 z a n α β z h0. 2 α β
24 M. Aydoğan, E. Yavuz Duman Sh. Owa Furthermore, we have that h z a + z n a n n=2 a + α β h0 2 α β a z h z a z n a n n=2 a α β h0 2 α β a z. Next, we consider Theorem 2.5. Let fz =zhzgz,where hz is given by 2. a n = a n e inθ+π θ R. Iffz satisfies zg 2.8 β 2 > max z U Re z > 0 gz 2.9 n + α β 2 a n α β 2 h0, then fz H LH α, where 0 <β 2 < α. Proof. Note that if fz satisfies 2.0 zh z hz < α β 2 z U, then we have that zh z Re >α+ β 2 hz z U. This implies that Re + zh z hz zg z >α gz z U.
Notes on starlike log-harmonic functions 25 Therefore, if fz satisfies the inequality 2.0, then fz HLH α. Indeed we see that zh z hz = n a n e inθ z n h0 a n e inθ z n < n a n h0 a n. Thus, if fz satisfies 2.9, then we have the inequality 2.0. 3 Coefficient Inequalities for gz Let fz =zhzgz be in the class SLH α. If fz satisfies 3. gz =g0 + b n z n g0 > 0 with b n = b n e inθ θ R, then we say that fz G LH α. Theorem 3.. 3.2 γ > max z U Re then If fz =zhzgz G LH α with zh z > 0, hz 3.3 n +α γ b n α + γ g0. Proof. Note that if fz G LH α S LH α, then zg z Re < Re α + zh z gz hz z U, which implies that Re zg z < α + γ z U. gz
26 M. Aydoğan, E. Yavuz Duman Sh. Owa Therefore, we see that zg z Re gz =Re nb nz n g0 + b nz n =Re n b n e inθ z n g0 + b n e inθ z n < α + γ z U. Let us consider a point z such that z = z e iθ U. Then, we have that zg z Re = n b n z n gz g0 + b n z < α + γ n z U. Thus, letting z, we obtain that n +α γ b n α + γ g0. Example 3.2. If fz =zhzgz G LH α with then gz =g0 + It follows that fz satisfies hz = 2γ z 0 < max z U Re γ > 0 α + γ g0e inθ nn + n +α γ zn, zh z <γ. hz n +α γ b n = α + γ g0. Applying Theorem 3., we have the following result. Theorem 3.3. If fz =zhzgz with 3.2, then 3.4 α + γ z g0 gz + α + γ z g0 α γ α γ
Notes on starlike log-harmonic functions 27 3, 5 b α + γ g0 α γ b z g z b + α + γ g0 α γ b z for z U, where 0 <γ <α. Proof. Since b n α + γ g0 α γ n b n α + γ g0 α γ b n=2 for fz G LH α, we prove the inequalities 3.4 3.5. Finally, we derive Theorem 3.4. Let fz =zhzgz, where gz is given by 3. b n = b n e inθ θ R. Iffz satisfies zh 3.6 γ 2 < min z U Re z < 0 hz 3.7 n +α γ 2 b n α + γ 2 g0 then fz G LH α, where α <γ 2 < 0. Proof. Note that if fz satisfies zg z gz < α + γ 2 z U, then Re zg z < α + γ 2 α +Re gz zh z hz z U,
28 M. Aydoğan, E. Yavuz Duman Sh. Owa which shows that fz SLH α. It follows that 3.8 zg z gz = n b n e inθ z n g0 + b n e inθ z n < n b n g0 b n α + γ 2 if the inequality 3.7 holds true. Therefore, we see that fz G LH α. 4 Open questions We know that Jahangiri [5] has showed the coefficient inequality which is the necessary sufficient condition for harmonic convex functions fz of order α in U. There are many necessary sufficient inequalities for some classes of analytic functions in U. We hope we will discuss some necessary sufficient conditions for starlike log-harmonic functions fz inu. References [] Z. Abdulhadi, Close-to-starlike logharmonic mappings, Internat. J. Math. Math. Sci. 9996, 563 574. [2] Z. Abdulhadi, Typically real logharmonic mappings, Internat. J. Math. Math. Sci. 32002, 9. [3] Z. Abdulhadi W. Hengartner, Spirallike logharmonic mappings, Complex Variables Theory Appl. 9987, 2 30. [4] Z. Abdulhadi Y. A. Muhanna, Starlike log-harmonic mappings of order α, J. Inequl. Pure Appl. Math. 7, Article 232006, 6. [5] J. M. Jahangiri, Coefficient bounds univalance criteria for harmonic functions with negative coefficients, Ann. Univ. Mariae Curie-Sklodowska 52998, 57 66.
Notes on starlike log-harmonic functions 29 [6] Y. Polatoğlu E. Deniz, Janowski starlike log-harmonic univalent functions, Hacettepe J. Math. Statistics 382009, 45 49. Received: August, 202