New sufficient conditions, concerned with the coefficients of harmonic
functions $f(z)=h(z)+\bar{g(z)}$ in the open unit disk $\mathbb{U}$ normalized
by $f(0)=h(0)=h'(0)-1=0$, for $f(z)$ to be harmonic close-to-convex functions
are discussed. Furthermore, several illustrative examples and the image domains
of harmonic close-to-convex functions satisfying the obtained conditions are
enumerated.Comment: 11 pages, 6 figure

HAYAMI, TOSHIO

Abstract and Applied Analysis

English

arXiv

Toshio Hayami

CiteSeerX

COEFFICIENT CONDITIONS FOR HARMONIC CLOSE-TO-CONVEX FUNCTIONS

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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2012, Article ID 413965, 12 pagesdoi:10.1155/2012/413965Research ArticleCoefficient Conditions for HarmonicClose-to-Convex FunctionsToshio HayamiDepartment of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, JapanCorrespondence should be addressed to Toshio Hayami, ha ya to112@hotmail.comReceived 25 January 2012; Accepted 13 April 2012Academic Editor: Roman Simon HilscherCopyright q 2012 Toshio Hayami. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.New suﬃcient conditions, concerned with the coeﬃcients of harmonic functions fz  hzgzin the open unit disk U normalized by f0  h0  h′0 − 1  0, for fz to be harmonic close-to-convex functions are discussed. Furthermore, several illustrative examples and the image domainsof harmonic close-to-convex functions satisfying the obtained conditions are enumerated.1. IntroductionFor a continuous complex-valued function fz  ux, y  ivx, y z  x  iy, we say thatfz is harmonic in the open unit disk U  {z ∈ C : |z| < 1} if both ux, y and vx, y are realharmonic in U, that is, ux, y and vx, y satisfy the Laplace equationsΔu  uxx  uyy  0, Δv  vxx  vyy  0. 1.1A complex-valued harmonic function fz in U is given by fz  hzgzwhere hz andgz are analytic in U. We call hz and gz the analytic part and the coanalytic part of fz,respectively. A necessary and suﬃcient condition for fz to be locally univalent and sensepreserving in U is |h′z| > |g ′z| in U see 1	 or 2	. Let H denote the class of harmonicfunctions fz in U with f0  h0  0 and h′0  1. Thus, every normalized harmonicfunction fz can be written byfz  hz  gz  z ∞∑n2anzn ∞∑n1bnzn ∈ H, 1.2where a1  1 and b0  0, for convenience.2 Abstract and Applied AnalysisWe next denote by SH the class of functions fz ∈ H that are univalent and sensepreserving in U. Due to the sense-preserving property of fz, we see that |b1|  |g ′0| <|h′0|  1. If gz ≡ 0, then SH reduces to the class S consisting of normalized analyticunivalent functions. Furthermore, for every function fz ∈ SH, the functionFz fz − b1fz1 − |b1|2 z ∞∑n2an − b1bn1 − |b1|2zn ∞∑n2bn − b1an1 − |b1|2zn 1.3is also a member of SH. Therefore, we consider the subclass S0H of SH defined asS0H {fz ∈ SH : b1  g ′0  0}. 1.4Conversely, if Fz ∈ S0H, then fz  Fz  b1Fz ∈ SH for any b1 |b1| < 1.We say that a domain D is a close-to-convex domain if the complement of D can bewritten as a union of nonintersecting half-lines except that the origin of one half-line may lieon one of the other half-lines. Let C, CH, and C0H be the respective subclasses of S, SH, andS0H consisting of all functions fz, which map U onto a certain close-to-convex domain.Bshouty and Lyzzaik 3	 have stated the following result.Theorem 1.1. If fz  hz  gz ∈ H satisfiesg ′z  zh′z, Re(1 zh′′zh′z)> −121.5for all z ∈ U, then fz ∈ C0H ⊂ S0H.A simple and interesting example is below.Example 1.2. The functionfz 1 − 1 − z221 − z2z221 − z2 z ∞∑n2n  12zn ∞∑n2n − 12zn 1.6satisfies the conditions of Theorem 1.1, and therefore fz belongs to the class C0H. We nowshow that fU is actually a close-to-convex domain. It follows thatfz (z21 − z2z21 − z)(z21 − z2− z21 − z) Re(z1 − z2) i Im(z1 − z).1.7Abstract and Applied Analysis 3Settingf(reiθ)−2r2  r(1  r2) cos θ1  r2 − 2r cos θ2r sin θ1  r2 − 2r cos θ i  u  iv 1.8for any z  reiθ ∈ U 0  r < 1, 0  θ < 2π, we see that−4(u  v2)4rr − cos θ1 − r cos θ1  r2 − 2r cos θ24rr − t1 − rt1  r2 − 2rt2≡ φt −1  t  cos θ  1.1.9Sinceφ′t −4r(1 − r2)21  r2 − 2rt3 0, 1.10we obtain thatφt  φ−1  4r1  r2≡ ψr. 1.11Also, noting thatψ ′r 41 − r1  r3> 0, 1.12we know thatψr < ψ1  1, 1.13which implies thatu > −v2 − 14. 1.14Thus, fzmaps U onto the following close-to-convex domain as shown in Figure 1.Remark 1.3. LetM be the class of all functions satisfying the conditions of Theorem 1.1. Then,it was earlier conjectured by Mocanu 4, 5	 that M ⊂ S0H. Furthermore, we can immediatelysee that the function fz in Example 1.2 is a member of the classM and it shows that fz ∈M is not necessarily starlike with respect to the origin in U fz is starlike with respect tothe origin in U if and only if tw ∈ fU for all w ∈ fU and t 0  t  1.4 Abstract and Applied Analysis0.5 10.511.5−0.5−0.5−1−1−1.5Figure 1: The image of fz  1 − 1 − z2/21 − z2  z2/21 − z2.Remark 1.4. For the function fz  hz  gz ∈ H given byg ′z  zn−1h′z n  2, 3, 4, . . ., 1.15letting wt  feit  heit  geit −π  t < π, we know thatIm(w′′tw′t) 0 −π  t < π, 1.16which means that fz maps the unit circle ∂U  {z ∈ C : |z|  1} onto a union of severalconcave curves see 6, Theorem 2.1	.Jahangiri and Silverman 7	 have given the following coeﬃcient inequality for fz ∈H to be in the class CH.Theorem 1.5. If fz ∈ H satisfies∞∑n2n | an | ∞∑n1n | bn | 1, 1.17then fz ∈ CH.Example 1.6. The functionfz  z 15z5 1.18Abstract and Applied Analysis 50.5 10.51−0.5−0.5−1−1Figure 2: The image of fz  z  1/5z5.belongs to the class C0H ⊂ CH and satisfies the condition of Theorem 1.5. Indeed, fzmaps Uonto the following hypocycloid of six cusps cf. 8	 or 6	 as shown in Figure 2.The object of this paper is to find some suﬃcient conditions for functions fz ∈ Hto be in the class CH. In order to establish our results, we have to recall here the followinglemmas due to Clunie and Sheil-Small 1	.Lemma 1.7. If hz and gz are analytic in U with |h′0| > |g ′0| and hz  εgz is close-to-convex for each ε |ε|  1, then fz  hz  gz is harmonic close-to-convex.Lemma 1.8. If fz  hz  gz is locally univalent in U and hz  εgz is convex for someε |ε|  1, then fz is univalent close-to-convex.We also need the following result due to Hayami et al. 9	.Lemma 1.9. If a function Fz  z ∑∞n2Anzn is analytic in U and satisfies∞∑n2⎡⎣∣∣∣∣∣∣n∑k1⎧⎨⎩k∑j1−1k−j j(j  1)(αk − j)Aj⎫⎬⎭(βn − k)∣∣∣∣∣∣∣∣∣∣∣∣n∑k1⎧⎨⎩k∑j1−1k−j j(j − 1)(αk − j)Aj⎫⎬⎭(βn − k)∣∣∣∣∣∣⎤⎦  21.19for some real numbers α and β, then Fz is convex in U.2. Main ResultsOur first result is contained in the following theorem.6 Abstract and Applied AnalysisTheorem 2.1. If fz ∈ H satisfies the following condition∞∑n2∣∣∣nan − eiϕn − 1an−1∣∣∣ ∞∑n1∣∣∣nbn − eiϕn − 1bn−1∣∣∣  1 2.1for some real number ϕ 0  ϕ < 2π, then fz ∈ CH.Proof. Let Fz  z ∑∞n2Anzn be analytic in U. If Fz satisfies∞∑n2∣∣∣nAn − eiϕn − 1An−1∣∣∣  1, 2.2then it follows that∣∣∣(1 − eiϕz)F ′z − 1∣∣∣ ∣∣∣∣∣∞∑n2(nAn − eiϕn − 1An−1)zn−1∣∣∣∣∣∞∑n2∣∣∣nAn − eiϕn − 1An−1∣∣∣ · | z |n−1<∞∑n2∣∣∣nAn − eiϕn − 1An−1∣∣∣  1 z ∈ U.2.3This gives us thatRe((1 − eiϕz)F ′z)> 0 z ∈ U, 2.4that is, Fz ∈ C. Then, it is suﬃcient to prove thatFz hz  εgz1  εb1 z ∞∑n2an  εbn1  εb1zn ∈ C 2.5for each ε |ε|  1 by Lemma 1.7. From the assumption of the theorem, we obtain that∞∑n2∣∣∣∣nan  εbn1  εb1− eiϕn − 1an−1  εbn−11  εb1∣∣∣∣ 11− | b1 |∞∑n2[∣∣∣nan − eiϕn − 1an−1∣∣∣ ∣∣∣nbn − eiϕn − 1bn−1∣∣∣] 1− | b1 |1− | b1 |  1.2.6This completes the proof of the theorem.Abstract and Applied Analysis 70.25 0.5 0.75 10.51−0.25−0.5−0.5−1Figure 3: The image of fz  −z − 2 log |1 − z|.Example 2.2. The functionfz  − log1 − z  (−mz − log1 − z)  z ∞∑n21nzn  1 −mz ∞∑n21nzn 0 < m  12.7satisfies the condition of Theorem 2.1 with ϕ  0 and belongs to the class CH. In particular,puttingm  1, we obtain Figure 3.By making use of Lemma 1.8 with ε  0 and applying Lemma 1.9, we readily obtainthe next theorem.Theorem 2.3. If fz ∈ H is locally univalent in U and satisfies∞∑n2⎡⎣∣∣∣∣∣∣n∑k1⎧⎨⎩k∑j1−1k−j j(j  1)(αk − j)aj⎫⎬⎭(βn − k)∣∣∣∣∣∣∣∣∣∣∣∣n∑k1⎧⎨⎩k∑j1−1k−j j(j − 1)(αk − j)aj⎫⎬⎭(βn − k)∣∣∣∣∣∣⎤⎦  22.8for some real numbers α and β, then fz ∈ CH.Putting α  β  0 in the above theorem, we arrive at the following result due toJahangiri and Silverman 7	.8 Abstract and Applied AnalysisTheorem 2.4. If fz ∈ H is locally univalent in U with∞∑n2n2|an|  1, 2.9then fz ∈ CH.Furthermore, taking α  1 and β  0 in the theorem, we have the following corollary.Corollary 2.5. If fz ∈ H is locally univalent in U and satisfies∞∑n2{n|n  1an − n − 1an−1|  n − 1|nan − n − 2an−1|}  2, 2.10then fz ∈ CH.Example 2.6. The functionfz  −∫z0log1 − ttdt (z  1 − z log1 − z)  z ∞∑n21n2zn ∞∑n21nn − 1zn 2.11satisfies the conditions of Corollary 2.5 and belongs to the class CH as shown in Figure 4.3. AppendixA sequence {cn}∞n0 of nonnegative real numbers is called a convex null sequence if cn → 0as n → ∞ andcn − cn1  cn1 − cn2  0 3.1for all n n  0, 1, 2, . . ..The next lemma was obtained by Feje´r 10	.Lemma 3.1. Let {cn}∞k0 be a convex null sequence. Then, the functionpz c02∞∑n1cnzn 3.2is analytic and satisfies Repz~~ > 0 in U.Applying the above lemma, we deduce the following theorem.Abstract and Applied Analysis 90.5 10.51−0.5−0.5−1Figure 4: The image of fz  − ∫z0 log1 − t/t dt  z  1 − z log1 − z~~.Theorem 3.2. For some b |b| < 1 and some convex null sequence {cn}∞n0 with c0  2, the functionfz  hz  gz  z ∞∑n2cn−1nzn  b(z ∞∑n2cn−1nzn)3.3belongs to the class CH.Proof. Let us define Fz byFz hz  εgz1  εb z ∞∑n2cn−1nzn 3.4for each ε |ε|  1. Then, we know thatF ′z c02∞∑n1cnzn c0  2. 3.5By virtue of Lemmas 1.7 and 3.1, it follows that ReF ′z > 0 z ∈ U, that is, Fz ∈ C. Thus,we conclude that fz  hz  gz ∈ CH.In the same manner, we also have the following theorem.10 Abstract and Applied AnalysisTheorem 3.3. For some b |b| < 1 and some convex null sequence {cn}∞n0 with c0  2, the functionfz  hz  gz  z ∞∑n21n⎛⎝1 n−1∑j1cj⎞⎠zn  b⎛⎝z ∞∑n21n⎛⎝1 n−1∑j1cj⎞⎠zn⎞⎠ 3.6belongs to the class CH.Proof. Let us define Fz byFz hz  εgz1  εb z ∞∑n21n⎛⎝1 n−1∑j1cj⎞⎠zn 3.7for each ε |ε|  1. Then, we know that1 − zF ′z  c02∞∑n1cnzn c0  2. 3.8Therefore, by the help of Lemmas 1.7 and 3.1, we obtain that Re1 − zF ′z > 0 z ∈ U,that is, Fz ∈ C, which implies that fz  hz  gz ∈ CH.Remark 3.4. The sequence{cn}∞n0 {2, 1,23, . . . ,2n  1, . . .}3.9is a convex null sequence becauselimn→∞cn  limn→∞(2n  1) 0, cn − cn1  2n  1n  2  0,cn − cn1 − cn1 − cn2  4n  1n  2n  3  0 n  0, 1, 2, . . ..3.10Setting b  1/4 in Theorem 3.2 with the above sequence {cn}∞n0, we derive thefollowing example.Example 3.5. The functionfz  −z − 2∫z0log1 − ttdt − 14(z  2∫z0log1 − ttdt) z ∞∑n22n2zn 14(z ∞∑n22n2zn)3.11is in the class CH as shown in Figure 5.Abstract and Applied Analysis 110.5 1 1.5 2 2.50.250.50.75−0.25−0.5−0.5−0.75Figure 5: The image of fz in Example 3.5.0.5 1 1.5 2 2.5 30.51−0.5−0.5−1Figure 6: The image of fz in Example 3.7.Moreover, we know the following remark.Remark 3.6. The sequence{cn}∞n0 {2, 1,12, . . . , 21−n, . . .}3.12is a convex null sequence becauselimn→∞cn  limn→∞21−n  0, cn − cn1  2−n  0,cn − cn1 − cn1 − cn2  2−n1  0 n  0, 1, 2, . . ..3.13Hence, letting b  1/4 in Theorem 3.3 with the sequence {cn}∞n0  {21−n}∞n0, we havethe following example.12 Abstract and Applied AnalysisExample 3.7. The functionfz  −3 log1 − z  4 log(1 − z2)(−34log1 − z  log(1 − z2)) z ∞∑n21n⎛⎝1 n−1∑j121−j⎞⎠zn 14⎛⎝z ∞∑n21n⎛⎝1 n−1∑j121−j⎞⎠zn⎞⎠3.14is in the class CH as shown in Figure 6.DedicationThis paper is dedicated to Professor Owa on the occasion of his retirement from KinkiUniversity.AcknowledgmentThe author expresses his sincere thanks to the referees for their valuable suggestions andcomments for improving this paper.References1	 J. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae.Series A, vol. 9, pp. 3–25, 1984.2	 H. Lewy, “On the non-vanishing of the Jacobian in certain one-to-one mappings,” Bulletin of theAmerican Mathematical Society, vol. 42, no. 10, pp. 689–692, 1936.3	 D. Bshouty and A. Lyzzaik, “Close-to-convexity criteria for planar harmonic mappings,” ComplexAnalysis and Operator Theory, vol. 5, no. 3, pp. 767–774, 2011.4	 P. T. Mocanu, “Three-cornered hat harmonic functions,” Complex Variables and Elliptic Equations, vol.54, no. 12, pp. 1079–1084, 2009.5	 P. T. Mocanu, “Injectivity conditions in the complex plane,” Complex Analysis and Operator Theory, vol.5, no. 3, pp. 759–766, 2011.6	 T. Hayami and S. Owa, “Hypocycloid of n  1 cusps harmonic function,” Bulletin of MathematicalAnalysis and Applications, vol. 3, pp. 239–246, 2011.7	 J. M. Jahangiri and H. Silverman, “Harmonic close-to-convex mappings,” Journal of AppliedMathematics and Stochastic Analysis, vol. 15, no. 1, pp. 23–28, 2002.8	 P. Duren, Harmonic Mappings in the Plane, vol. 156 of Cambridge Tracts in Mathematics, CambridgeUniversity Press, Cambridge, UK, 2004.9	 T. Hayami, S. Owa, and H. M. Srivastava, “Coeﬃcient inequalities for certain classes of analytic andunivalent functions,” Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 4, article 95, pp.1–10, 2007.10	 L. Feje´r, “U¨ber die positivita¨t von summen, die nach trigonometrischen oder Legendreschen, funktio-nen fortschreiten. I,” Acta Szeged, vol. 2, pp. 75–86, 1925.Submit your manuscripts athttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical Problems in EngineeringHindawi Publishing Corporationhttp://www.hindawi.comDifferential EquationsInternational Journal ofVolume 2014Applied MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical PhysicsAdvances inComplex AnalysisJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014OptimizationJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Operations ResearchAdvances inJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Function SpacesAbstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal of Mathematics and Mathematical SciencesHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AlgebraDiscrete Dynamics in Nature and SocietyHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Decision SciencesAdvances inDiscrete MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Stochastic AnalysisInternational Journal of

Coefficient Conditions for Harmonic Close-to-Convex Functions

New sufficient conditions, concerned with the coefficients of harmonic functions ()=ℎ()+() in the open unit disk  normalized by (0)=ℎ(0)=ℎ(0)−1=0, for () to be harmonic close-to-convex functions are discussed. Furthermore, several illustrative examples and the image domains of harmonic close-to-convex functions satisfying the obtained conditions are enumerated

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ADV  MATH
SCI  JOURNAL
Advances in Mathematics: Scientic Journal 1 (2012), no.2, 95100
ISSN 1857-8365, UDC: 517.546:517.57
COEFFICIENT CONDITIONS FOR HARMONIC
CLOSE-TO-CONVEX FUNCTIONS
TOSHIO HAYAMI
Presented at the 8th International Symposium GEOMETRIC FUNCTION THEORY AND
APPLICATIONS, 27-31 August 2012, Ohrid, Republic of Macedonia.
Abstract. New sucient conditions, concerned with the coecients of harmonic
functions f(z) = h(z)+g(z) in the open unit disk normalized by f(0) = h(0) = h0(0) 
1 = 0, for f(z) to be harmonic close-to-convex functions are discussed. Furthermore,
several illustrative examples of the obtained theorems are enumerated.
1. Introduction
Let D be a simply connected domain in the complex plane C. A continuous complex-
valued function f(z) = u(x; y) + iv(x; y) in D (z = x + iy 2 D), we say that f(z) is
harmonic in D if both u(x; y) and v(x; y) are real harmonic in D, that is, u(x; y) and
v(x; y) satisfy the Laplace's equations
u = uxx + uyy = 0 and v = vxx + vyy = 0 (z = x+ iy 2 D):
A harmonic function f(z) in D is given by f(z) = h(z) + g(z) where h(z) and g(z)
are analytic in D. We call h(z) and g(z) the analytic part and the co-analytic part
of f(z), respectively. The Jacobian of f(z) denoted by Jf can be computed by Jf =
jh0(z)j2 jg0(z)j2. A necessary and sucient condition for f(z) to be locally univalent and
sense-preserving in D is that Jf > 0, that is, that jg
0(z)j < jh0(z)j in D (see, [2] or [9]). Let
H denote the class of harmonic functions f(z) in the open unit disk U = fz 2 C : jzj < 1g
with f(0) = h(0) = 0 and h0(0) = 1. Thus, all functions f(z) 2 H can be written by
f(z) = h(z) + g(z) = z +
1X
n=2
anz
n +
1X
n=1
bnzn
where a1 = 1 and b0 = 0, for convenience.
2010 Mathematics Subject Classication. 30C45, 58E20.
Key words and phrases. Coecient conditions, harmonic functions, univalent functions, close-to-
convex functions.
95
96 T. HAYAMI
We next denote by SH the class of functions f(z) 2 H which are univalent and sense-
preserving in U. Since the sense-preserving property of f(z), we see that jb1j = jg
0(0)j <
jh0(0)j = 1. If g(z)  0, then SH reduces to the class S consisting of normalized analytic
univalent functions. Furthermore, for every function f(z) 2 SH, the function
F (z) =
f(z)  b1f(z)
1  jb1j2
= z +
1X
n=2
an   b1bn
1  jb1j2
zn +
1X
n=2
bn   b1an
1  jb1j2
zn
is also a member of SH. Therefore, we consider the subclass S
0
H
of SH dened as
S0H = ff(z) 2 SH : b1 = g
0(0) = 0g :
Conversely, if F (z) 2 S0
H
, then f(z) = F (z) + b1F (z) 2 SH for any b1 (jb1j < 1).
We say that D is a close-to-convex domain if the complement of D can be written as a
union of non-intersecting half-lines (except that the origin of one half-line may lie on one
of the other half-lines). Let C, CH and C
0
H
be the respective subclasses of S, SH and S
0
H
consisting of all functions f(z) which map U onto a certain close-to-convex domain.
A simple and interesting example is below.
Example 1.1. The function
f(z) =
1  (1  z)2
2(1  z)2
+
z2
2(1  z)2
= z +
1X
n=2
n+ 1
2
zn +
1X
n=2
n  1
2
zn
belongs to the class C0
H
.
The aim of this paper is to nd new sucient conditions for functions f(z) 2 H to be
in the class CH. To obtain our results, we have to recall here the following lemmas due
to Clunie and Sheil-small [2].
Lemma 1.1. If h(z) and g(z) are analytic in U with jh0(0)j > jg0(0)j and h(z)+ "g(z)
is close-to-convex for each " (j"j = 1), then f(z) = h(z) + g(z) is harmonic close-to-
convex.
Lemma 1.2. If f(z) = h(z)+g(z) is locally univalent in U and h(z)+"g(z) is convex
for some " (j"j  1), then f(z) is univalent close-to-convex.
We also need the following result due to Hayami, Owa and Srivastava [5].
Lemma 1.3. If a function F (z) = z +
1P
n=2
Anz
n is analytic in U and satises
1X
n=2
2
4

nX
k=1
8<
:
kX
j=1
( 1)k jj(j + 1)


k   j

Aj
9=
;


n  k

+

nX
k=1
8<
:
kX
j=1
( 1)k jj(j   1)


k   j

Aj
9=
;


n  k

3
5  2
COEFFICIENT CONDITIONS FOR HARMONIC . . . 97
for some real numbers  and , then F (z) is convex in U.
2. Main results
Using Lemma 1.1 we receive
Theorem 2.1. If f(z) 2 H satises the following condition
1X
n=2
nan   ei'(n  1)an 1+ 1X
n=1
nbn   ei'(n  1)bn 1  1
for some real number ' (0  ' < 2), then f(z) 2 CH.
Example 2.1. The function
f(z) =   log(1 z)+

 mz   log(1  z)

= z+
1X
n=2
1
n
zn+(1 m)z+
1X
n=2
1
n
zn (0 < m  1)
satises the condition of Theorem 2.1 with ' = 0 and belongs to the class CH.
By making use of Lemma 1.2 with " = 0 and applying Lemma 1.3, we readily obtain
the following theorem.
Theorem 2.2. If f(z) 2 H is locally univalent in U and satises
1X
n=2
2
4

nX
k=1
8<
:
kX
j=1
( 1)k jj(j + 1)


k   j

aj
9=
;


n  k

+

nX
k=1
8<
:
kX
j=1
( 1)k jj(j   1)


k   j

aj
9=
;


n  k

3
5  2
for some real numbers  and , then f(z) 2 CH.
Putting  =  = 0 in the above theorem, we arrive at the following result due to
Jahangiri and Silverman [8].
Corollary 2.1. If f(z) 2 H is locally univalent in U with
1X
n=2
n2janj  1;
then f(z) 2 CH.
Furthermore, taking  = 1 and  = 0 in the theorem, we have
98 T. HAYAMI
Corollary 2.2. If f(z) 2 H is locally univalent in U and satises
1X
n=2
fn j(n+ 1)an   (n  1)an 1j+ (n  1) jnan   (n  2)an 1jg  2;
then f(z) 2 CH.
Example 2.2. The function
f(z) =  
Z z
0
log(1  t)
t
dt+

z + (1  z) log(1  z)

= z +
1X
n=2
1
n2
zn +
1X
n=2
1
n(n  1)
zn
satises the conditions of Corollary 2.2 and belongs to the class CH.
3. Appendix
A sequence fcng
1
n=0 of non-negative real numbers is called a convex null sequence if
cn ! 0 as n!1 and
cn   cn+1  cn+1   cn+2  0
for all n (n = 0; 1; 2;    ).
The next lemma was obtained by Fejér [4].
Lemma 3.1. Let fcng
1
n=0 be a convex null sequence. Then, the function
p(z) =
c0
2
+
1X
n=1
cnz
n
is analytic and satises Re(p(z)) > 0 in U.
Applying the above lemma, we deduce
Theorem 3.1. For some b (jbj < 1) and some convex null sequence fcng
1
n=0 with
c0 = 2, the function
f(z) = h(z) + g(z) = z +
1X
n=2
cn 1
n
zn + b
 
z +
1X
n=2
cn 1
n
zn
!
belongs to the class CH.
In the same manner, we also have
Theorem 3.2. For some b (jbj < 1) and some convex null sequence fcng
1
n=0 with
c0 = 2, the function
f(z) = h(z) + g(z) = z +
1X
n=2
1
n
0
@1 + n 1X
j=1
cj
1
A zn + b
0
@z + 1X
n=2
1
n
0
@1 + n 1X
j=1
cj
1
A zn
1
A
belongs to the class CH.
COEFFICIENT CONDITIONS FOR HARMONIC . . . 99
Remark 3.1. The sequence
fcng
1
n=0 =

2; 1;
1
2
;    ; 21 n;   

is a convex null sequence because
lim
n!1
cn = lim
n!1
21 n = 0; cn   cn+1 = 2
 n  0
and
(cn   cn+1)  (cn+1   cn+2) = 2
 (n+1)  0 (n = 0; 1; 2;    ):
According to Remark 3.1, letting b = 1=4 in Theorem 3.2 with the sequence fcng
1
n=0 =
f21 ng1n=0, we have
Example 3.1. The function
f(z) =  3 log(1  z) + 4 log

1 
z
2

+

 
3
4
log(1  z) + log

1 
z
2

= z +
1X
n=2
1
n
0
@1 + n 1X
j=1
21 j
1
A zn + 1
4
0
@z + 1X
n=2
1
n
0
@1 + n 1X
j=1
21 j
1
A zn
1
A
is in the class CH.
The details of this article can be found in the paper [7].
References
[1] D. Bshouty, A. Lyzzaik: Close-to-convexity criteria for planar harmonic mappings, Complex
Anal. Oper. Theory 5(2011), 767774.
[2] J. Clunie, T. Sheil-Small: Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math.
9 (1984), 325.
[3] P. L. Duren: Harmonic Mappings in the Plane, Cambridge University Press, Cambridge, 2004.
[4] L. Fejér: Über die positivität von summen, die nach trigonometrischen oder Legendreschen, funk-
tionen fortschreiten. I, Acta Szeged 2 (1925), 75-86.
[5] T. Hayami, S. Owa, H. M. Srivastava: Coecient inequalities for certain classes of analytic
and univalent functions, J. Inequal. Pure Appl. Math. 8 (2007), Article 95, 10 pp.
[6] T. Hayami, S. Owa: Hypocycloid of n + 1 cusps harmonic function, Bull. Math. Anal. Appl. 3
(2011), 239246.
[7] T. Hayami: Coecient conditions for harmonic close-to-convex functions, Abstr. Appl. Anal.
(2012), Article ID 413965, 12 pp.
[8] J. M. Jahangiri, H. Silverman: Harmonic close-to-convex mappings, J. Appl. Math. Stoch.
Anal. 15(2002), 2328.
[9] H. Lewy: On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer.
Math. Soc. 42 (1936), 689692.
[10] P. T. Mocanu: Three-cornered hat harmonic functions, Complex Var. Elliptic Equ. 54 (2009),
10791084.
[11] P. T. Mocanu: Injectively conditions in the complex plane, Complex Anal. Oper. Theory 5 (2011),
759766.
100 T. HAYAMI
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577-8502, Japan
E-mail address: ha_ya_to112@hotmail.com


COEFFICIENT CONDITIONS FOR HARMONIC CLOSE-TO-CONVEX FUNCTIONS

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