For log-harmonic functions f(z) = zh(z)g(z) in the open unit disk U, two subclasses H*LH(?) and G*LH(?) of S*LH(?) consisting of all starlike log-harmonic functions of order ? (0 ? ? &lt; 1) are considered. The object of the present paper is to discuss some coefficient inequalities for h(z) and g(z).Mathematics Subject Classification: Primary 30C55, Secondary 30C45.Publisher's Versio

Aydoğan, Seher Melike

Duman, Emel Yavuz

Owa, Shigeyoshi

International Journal of Mathematical Analysis

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Isik University Academic Open Access

Int. Journal of Math. Analysis, Vol. 7, 2013, no. 1, 19 - 29Notes on Starlike log-Harmonic Functionsof Order αMelike AydoğanDepartment of MathematicsIşık University, Meşrutiyet KoyuŞile İstanbul, Turkeymelike.aydogan@isikun.edu.trEmel Yavuz DumanDepartment of Mathematics and Computer Scienceİstanbul Kültür Universityİstanbul, Turkeye.yavuz@iku.edu.trShigeyoshi OwaDepartment of MathematicsKinki UniversityHigashi-Osaka, Osaka 577-8502, Japanshige21@ican.zaq.ne.jpAbstractFor log-harmonic functions f(z) = zh(z)g(z) in the open unit disk U,two subclasses H∗LH(α) and G∗LH(α) of S∗LH(α) consisting of all starlikelog-harmonic functions of order α (0 ≤ α < 1) are considered. Theobject of the present paper is to discuss some coefficient inequalities forh(z) and g(z).20 M. Aydoğan, E. Yavuz Duman and Sh. OwaMathematics Subject Classification: Primary 30C55, Secondary 30C45Keywords: Analytic, log-harmonic, starlike, coefficient inequality.1 IntroductionLet H be the class of functions which are analytic in the open unit discU = {z ∈ C : |z| < 1}. A log-harmonic function f(z) is a solution of thenon-linear elliptic partial differential equation(1.1)fzf= w(z)fzf,where w(z) ∈ H satisfies |w(z)| < 1 (z ∈ U) and is said to be the seconddilatation, and(1.2) fz =12(∂f∂x− i∂f∂y), fz =12(∂f∂x+ i∂f∂y).Let a function f(z) given by(1.3) f(z) = zh(z)g(z)with 0 /∈ hg(U) be log-harmonic function in U, where h(z) ∈ H and g(z) ∈ H .Then f(z) is said to be starlike log-harmonic function of order α if it satisfies(1.4)∂(arg f(reiθ))∂θ= Re(zfz − zfzf)> α (z ∈ U)for some real α (0 ≤ α < 1). We denote by S∗LH(α) all starlike log-harmonicfunctions f(z) of order α in U.The class S∗LH(α) was studied by Abdulhadi and Muhanna [4], and Po-latoğlu and Deniz [6]. Furthermore, the classes of univalent log-harmonic func-tions have been studied by Abdulhadi [1], [2], and Abdulhadi and Hengartner[3].Notes on starlike log-harmonic functions 212 Coefficient Inequalities for h(z)In order to consider our problem, we have to introduce the following subclassH∗LH(α) of S∗LH(α). A function f(z) = zh(z)g(z) ∈ S∗LH(α) is said to be in aclass H∗LH(α) if it satisfies(2.1) h(z) = h(0) +∞∑n=1anzn (h(0) > 0)with an = |an| ei(nθ+π) (θ ∈ R).Now we deriveTheorem 2.1. If f = zh(z)g(z) ∈ H∗LH(α) with(2.2) β1 < minz∈ Re(zg′(z)g(z))< 0,then(2.3)∞∑n=1(n + 1 − α − β1) |an| ≤ (1 − α − β1)h(0).Proof. Note that f = zh(z)g(z) ∈ H∗LH(α) ⊂ S∗LH(α) satisfies∂∂θ(arg f(reiθ)) = Re(zfz − zfzf)= Re(1 +zh′(z)h(z)− zg′(z)g(z))> α (z ∈ U).This gives us that(2.4) Re(zh′(z)h(z))= Re( ∑∞n=1 nanznh(0) +∑∞n=1 anzn)= Re( −∑∞n=1 n |an| einθznh(0) −∑∞n=1 |an| einθzn)> Re(α − 1 + zg′(z)g(z))22 M. Aydoğan, E. Yavuz Duman and Sh. Owa> α + β1 − 1for all z ∈ U.Let us consider a point z such that z = |z| e−iθ ∈ U. Then (2.4) becomesthat(2.5) Re(zh′(z)h(z))=−∑∞n=1 n|an||z|nh(0) −∑∞n=1 |an||z|n > α + β1 − 1 (z ∈ U).Letting |z| → 1−, we obtain that−∞∑n=1n |an| ≥ (α + β1 − 1)(h(0) −∞∑n=1|an|),that is, that∞∑n=1(n + 1 − α − β1) |an| ≤ (1 − α − β1)h(0).Example 2.2. Let us consider a function f(z) = zh(z)g(z) ∈ H∗LH(α)withh(z) = h(0) +∞∑n=1(1 − α − β1)h(0)einθn(n + 1)(n + 1 − α − β1)znandg(z) =2β11 − z (β1 < 0).Then0 > minz∈ Re(zg′(z)g(z))> β1and∞∑n=1(n + 1 − α − β1) |an| = (1 − α − β1)h(0)( ∞∑n=11n(n + 1))= (1 − α − β1)h(0).Theorem 2.1 gives us the following corollary.Corollary 2.3. If f(z) = zh(z)g(z) ∈ H∗LH(α) with (2.2) then|an| ≤ 1 − α − β1n + 1 − α − β1h(0) (n = 1, 2, 3, · · · ).Notes on starlike log-harmonic functions 23Next, we showTheorem 2.4. If f(z) = zh(z)g(z) ∈ H∗LH(α) with (2.2) then(2.6)(1 − 1 − α − β12 − α − β1 |z|)h(0) ≤ |h(z)| ≤(1 +1 − α − β12 − α − β1 |z|)h(0)and(2.7) |a1| − ((1 − α − β)h(0) − (2 − α − β) |a1|) |z|≤ |h′(z)| ≤ |a1| + ((1 − α − β)h(0) − (2 − α − β) |a1|) |z|for z ∈ U. The equality in (2.6) holds for f(z) = zh(z)g(z) withh(z) = h(0) +1 − α − β12 − α − β1h(0)eiθz.Proof. We note that the inequality (2.3) gives us that∞∑n=1|an| ≤ 1 − α − β12 − α − β1 h(0)and∞∑n=2n |an| ≤ (1 − α − β1)h(0) − (2 − α − β1) |a1| .Thus, we have that|h(z)| ≤ h(0) + |z|∞∑n=1|an| ≤(1 +1 − α − β12 − α − β1 |z|)h(0)and|h(z)| ≥ h(0) − |z|∞∑n=1|an| ≥(1 − 1 − α − β12 − α − β1 |z|)h(0).24 M. Aydoğan, E. Yavuz Duman and Sh. OwaFurthermore, we have that|h′(z)| ≤ |a1| + |z|∞∑n=2n |an|≤ |a1| + ((1 − α − β1)h(0) − (2 − α − β1) |a1|) |z|and|h′(z)| ≥ |a1| − |z|∞∑n=2n |an|≥ |a1| − ((1 − α − β1)h(0) − (2 − α − β1) |a1|) |z| .Next, we considerTheorem 2.5. Let f(z) = zh(z)g(z),where h(z) is given by (2.1) andan = |an| ei(nθ+π) (θ ∈ R). If f(z) satisfies(2.8) β2 > maxz∈ Re(zg′(z)g(z))> 0and(2.9)∞∑n=1(n + 1 − α − β2)|an| ≤ (1 − α − β2)h(0),then f(z) ∈ H∗LH(α), where 0 < β2 < 1 − α.Proof. Note that if f(z) satisfies(2.10)∣∣∣∣zh′(z)h(z)∣∣∣∣ < 1 − α − β2 (z ∈ U),then we have thatRe(zh′(z)h(z))> α + β2 − 1 (z ∈ U).This implies thatRe(1 +zh′(z)h(z)− zg′(z)g(z))> α (z ∈ U).Notes on starlike log-harmonic functions 25Therefore, if f(z) satisfies the inequality (2.10), then f(z) ∈ H∗LH(α). Indeedwe see that∣∣∣∣zh′(z)h(z)∣∣∣∣ =∣∣∣∣ −∑∞n=1 n |an| einθznh(0) −∑∞n=1 |an| einθzn∣∣∣∣ <∑∞n=1 n |an|h(0) −∑∞n=1 |an| .Thus, if f(z) satisfies (2.9), then we have the inequality (2.10).3 Coefficient Inequalities for g(z)Let f(z) = zh(z)g(z) be in the class S∗LH(α). If f(z) satisfies(3.1) g(z) = g(0) +∞∑n=1bnzn (g(0) > 0)with bn = |bn| einθ (θ ∈ R), then we say that f(z) ∈ G∗LH(α).Theorem 3.1. If f(z) = zh(z)g(z) ∈ G∗LH(α) with(3.2) γ1 > maxz∈ Re(zh′(z)h(z))> 0,then(3.3)∞∑n=1(n − 1 + α − γ1) |bn| ≤ (1 − α + γ1)g(0).Proof. Note that if f(z) ∈ G∗LH(α) ⊂ S∗LH(α), thenRe(zg′(z)g(z))< Re(1 − α + zh′(z)h(z))(z ∈ U),which implies thatRe(zg′(z)g(z))< 1 − α + γ1 (z ∈ U).26 M. Aydoğan, E. Yavuz Duman and Sh. OwaTherefore, we see thatRe(zg′(z)g(z))= Re( ∑∞n=1 nbnzng(0) +∑∞n=1 bnzn)= Re( ∑∞n=1 n |bn| einθzng(0) +∑∞n=1 |bn| einθzn)< 1 − α + γ1 (z ∈ U).Let us consider a point z such that z = |z| e−iθ ∈ U. Then, we have thatRe(zg′(z)g(z))=∑∞n=1 n |bn| |z|ng(0) +∑∞n=1 |bn| |z|n< 1 − α + γ1 (z ∈ U).Thus, letting |z| → 1−, we obtain that∞∑n=1(n − 1 + α − γ1) |bn| ≤ (1 − α + γ1)g(0).Example 3.2. If f(z) = zh(z)g(z) ∈ G∗LH(α) withh(z) =2γ11 − z (γ1 > 0)andg(z) = g(0) +∞∑n=1(1 − α + γ1)g(0)einθn(n + 1)(n − 1 + α − γ1)zn,then0 < maxz∈ Re(zh′(z)h(z))< γ1.It follows that f(z) satisfies∞∑n=1(n − 1 + α − γ1) |bn| = (1 − α + γ1)g(0).Applying Theorem 3.1, we have the following result.Theorem 3.3. If f(z) = zh(z)g(z) with (3.2), then(3.4)(1 − 1 − α + γ1α − γ1 |z|)g(0) ≤ |g(z)| ≤(1 +1 − α + γ1α − γ1 |z|)g(0)Notes on starlike log-harmonic functions 27and(3, 5) |b1| − ((1 − α + γ1)g(0) − (α − γ1) |b1|) |z|≤ |g′(z)| ≤ |b1| + ((1 − α + γ1)g(0) − (α − γ1) |b1|) |z|for z ∈ U, where 0 < γ1 < α.Proof. Since∞∑n=1|bn| ≤ 1 − α + γ1α − γ1 g(0)and ∞∑n=2n |bn| ≤ (1 − α + γ1)g(0) − (α − γ1) |b1|for f(z) ∈ G∗LH(α), we prove the inequalities (3.4) and (3.5).Finally, we deriveTheorem 3.4. Let f(z) = zh(z)g(z), where g(z) is given by (3.1) andbn = |bn| einθ (θ ∈ R). If f(z) satisfies(3.6) γ2 < minz∈ Re(zh′(z)h(z))< 0and(3.7)∞∑n=1(n − 1 + α − γ2) |bn| ≤ (1 − α + γ2)g(0)then f(z) ∈ G∗LH(α), where α − 1 < γ2 < 0.Proof. Note that if f(z) satisfies∣∣∣∣zg′(z)g(z)∣∣∣∣ < 1 − α + γ2 (z ∈ U),thenRe(zg′(z)g(z))< 1 − α + γ2 ≤ 1 − α + Re(zh′(z)h(z))(z ∈ U),28 M. Aydoğan, E. Yavuz Duman and Sh. Owawhich shows that f(z) ∈ S∗LH(α).It follows that(3.8)∣∣∣∣zg′(z)g(z)∣∣∣∣ =∣∣∣∣∑∞n=1 n |bn| einθzng(0) +∑∞n=1 |bn| einθzn∣∣∣∣<∑∞n=1 n |bn|g(0) −∑∞n=1 |bn| ≤ 1 − α + γ2if the inequality (3.7) holds true. Therefore, we see that f(z) ∈ G∗LH(α).4 Open questionsWe know that Jahangiri [5] has showed the coefficient inequality which isthe necessary and sufficient condition for harmonic convex functions f(z) oforder α in U. There are many necessary and sufficient inequalities for someclasses of analytic functions in U. We hope we will discuss some necessary andsufficient conditions for starlike log-harmonic functions f(z) in U.References[1] Z. Abdulhadi, Close-to-starlike logharmonic mappings, Internat. J. Math.Math. Sci. 19(1996), 563–574.[2] Z. Abdulhadi, Typically real logharmonic mappings, Internat. J. Math.Math. Sci. 31(2002), 1–9.[3] Z. Abdulhadi and W. Hengartner, Spirallike logharmonic mappings, Com-plex Variables Theory Appl. 9(1987), 121–130.[4] Z. Abdulhadi and Y. A. Muhanna, Starlike log-harmonic mappings oforder α, J. Inequl. Pure Appl. Math. 7, Article 123(2006), 1–6.[5] J. M. Jahangiri, Coefficient bounds and univalance criteria for harmonicfunctions with negative coefficients, Ann. Univ. Mariae Curie-Sklodowska52(1998), 57–66.Notes on starlike log-harmonic functions 29[6] Y. Polatoğlu and E. Deniz, Janowski starlike log-harmonic univalent func-tions, Hacettepe J. Math. Statistics 38(2009), 45–49.Received: August, 2012

Notes on starlike log-harmonic functions of order α

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Notes on starlike log-harmonic functions of order α

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