Notes on Starlike log-harmonic Functions. of Order α

Transcription

1 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 Emel Yavuz Duman Department of Mathematics Computer Science İstanbul Kültür University İstanbul, Turkey Shigeyoshi Owa Department of Mathematics Kinki University Higashi-Osaka, Osaka , Japan 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.

2 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].

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

4 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,.

5 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

6 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.

7 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 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 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

8 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

9 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 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,

10 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, [2] Z. Abdulhadi, Typically real logharmonic mappings, Internat. J. Math. Math. Sci , 9. [3] Z. Abdulhadi W. Hengartner, Spirallike logharmonic mappings, Complex Variables Theory Appl. 9987, [4] Z. Abdulhadi Y. A. Muhanna, Starlike log-harmonic mappings of order α, J. Inequl. Pure Appl. Math. 7, Article , 6. [5] J. M. Jahangiri, Coefficient bounds univalance criteria for harmonic functions with negative coefficients, Ann. Univ. Mariae Curie-Sklodowska 52998,

11 Notes on starlike log-harmonic functions 29 [6] Y. Polatoğlu E. Deniz, Janowski starlike log-harmonic univalent functions, Hacettepe J. Math. Statistics , Received: August, 202