MSC 2010: 30C55, 30C45Distortion and growth theorems are obtained

Yavuz Duman, Emel

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

SOME DISTORTION THEOREMS FORSTARLIKE HARMONIC FUNCTIONSEmel Yavuz DumanDedicated to Professor Yas¸ar Polatog˘luthe occasion of his 60th birthdayAbstractIn this paper, we consider harmonic univalent mappings of the formf = h + g¯ defined on the unit disk D which are starlike. Distortion andgrowth theorems are obtained.MSC 2010: 30C55, 30C45Key Words and Phrases: harmonic function, starlike harmonic function,growth theorem, distortion theorem1. IntroductionA continuous complex-valued function f = u + iv defined in a simplyconnected domain D is said to be harmonic in D if both u and v are realharmonic in D, that is, u, v satisfy, respectively the Laplace equations∆u = uxx + uyy = 0, ∆v = vxx + vyy = 0.There is a well-known relation between analytic functions and harmonicfunctions. For example, for real harmonic functions u and v which aredefined on a simply connected domain D there exist analytic functions Uand V so thatu = Re(U) and v = Im(V ).c© 2010, FCAA – Diogenes Co. (Bulgaria). All rights reserved.546 E. Yavuz DumanTherefore, it has a canonical decompositionf = h+ g (1)where h and g are, respectively, the analytic functionsh =U + V2and g =U − V2.We call h the analytic part and g the co-analytic part of f . It is factthat if f = u + iv has continuous partial derivatives, then f is analyticif and only if the Cauchy-Riemann equations are satisfied. It follows thatevery analytic function is a complex-valued harmonic function. However,not every complex-valued harmonic function is analytic.The Jacobian Jf of a function f = u+ iv has a very important place inthe theory of harmonic mappings, defined byJf =∣∣∣∣ ux uyvx vy∣∣∣∣ = uxvy − uyvx.Or, in terms of fz and fz¯, we haveJf = |fz|2 − |fz¯|2 = |h′(z)|2 − |g′(z)|2,where f = h+ g is the harmonic function in D.If f = h+ g is a harmonic function on D with Jf > 0, then we say thatf is a sense-preserving (or orientation preserving) harmonic function on D.In this case we have|g′(z)| < |h′(z)|for all z ∈ D. If f has Jf < 0, then f is sense preserving. For convenience,we will only examine sense preserving harmonic functions.The mapping z → f(z) is sense preserving and locally univalent in Dif and only if Jf > 0 in D. The function f = h + g¯ is said to be harmonicunivalent in D if the mapping z → f(z) is sense preserving harmonic andunivalent in D.The second complex dilatation of a harmonic function f = h+ g is thequantityω(z) =fz¯fz=g′(z)h′(z)(z ∈ D). (2)Let SH denote the family of functions f = h + g that are harmonic,sense preserving, and univalent in the open unit disc D := {z ∈ C : |z| < 1}with the normalizationSOME DISTORTION THEOREMS . . . 547h(z) = z +∞∑n=2anzn, g(z) =∞∑n=1bnzn. (3)It follows from the sense-preserving property if f ∈ SH, then we have|ω(z)| < 1 for all z ∈ D. Thus, it is easy to see that |b1| < 1. Since thesecond complex dilatation ω of a sense preserving harmonic mapping f isalways an analytic function of modulus less than one, then this functionω will be called the analytic dilatation of f . Also f ∈ SH reduces to theclass of normalized analytic univalent functions if the co-analytic part of itsmembers is zero. In 1984 Clunie and Sheil-Small [1] investigated the classSH as well as its geometric subclasses and obtained some coefficient bounds.Many studies have been done on this class and its subclasses, and continuedtaking place.A sense-preserving harmonic mapping f ∈ SH is in the class S∗H if therange f(D) is starlike with respect to the origin. A function f ∈ S∗H is acalled harmonic starlike mapping in D. A function f = h + g with such aproperty must satisfy the conditionRe(zh′(z)− zg′(z)h(z) + g(z))> 0for all z ∈ D.In our proofs we use the following lemma:Lemma 1.1[2]. If f = h+ g ∈ S∗H, then there exist angles α and β suchthatRe{(eiαh(z)z+ e−iαg(z)z)(eiβ − e−iβz2)}> 0 (4)for all z ∈ D.Let A denote the class of all functions s1 analytic in the open unit disk Dwith the usual normalization s1(0) = s′1(0)−1 = 0. If s1 and s2 are analyticin D, we say that s1 is subordinate to s2, written s1 ≺ s2 or s1(z) ≺ s2(z),if s2 is univalent, then we have s1(0) = s2(0) and s1(D) ⊂ s2(D).Let P be the class of functions p of the formp(z) = 1 +∞∑n=1pnzn548 E. Yavuz Dumanwhich are analytic in the open unit disk D. If p in P satisfies Rep(z) > 0 forz ∈ D, then we say that p is the Carathe´odory function. It has been shownthat for a function p(z) ∈ P, the following inequalities are satisfied ([3]):1− r1 + r≤ |p(z)| ≤ 1 + r1− r , (5)and ∣∣∣∣z p′(z)p(z)∣∣∣∣ ≤ 2r1− r2 (6)for all |z| = r < 1.2. ResultsLemma 2.1. Let f = h+ g be an element of S∗H, then we have(A+ |B|)r − (A− |B|)r2(1 + r)(1 + r2)≤ |h(z)− e−2iαg(z)|≤ (A+ |B|)r + (A− |B|)r2(1 + r)(1 + r2),(7)for |z| = r < 1 where A = cos(β + α) − a cos(β − α) + b sin(β − α) > 0,B = sin(β + α) − b cos(β − α) − a sin(β − α), g′(0) = b1 = a + ib for somechoice of angles α and β.P r o o f. Since f = h+ g is element of S∗H, then we haveh(z)z∣∣z=0= 1,g(z)z∣∣z=0= b1 = a+ ib,and if we consider (4) as a function with positive real partp(z) =(eiαh(z)z+ e−iαg(z)z)(eiβ − e−iβz2)(8)has the properties Rep(z) > 0 and p(0) = [cos(β + α) − a cos(β − α) +b sin(β−α)]i[sin(β+α)− b cos(β−α)− a sin(β−α)] where b1 = a+ ib andα, β are angles.On the other hand, the assumption p(0) = 1 is not restriction for theCarathe´dory class. Indeed, let p(z) be element of the Carathe´dory classwith p(0) = A+ iB, A > 0, then the functionp1(z) =1A(p(z)− iB)satisfies the condition p1(0) = 1 and Rep1(z) > 0. This shows that p1(z) isthe element of the Carathe´dory class. Therefore, the functionSOME DISTORTION THEOREMS . . . 549p1(z) =1A(p(z)− iB) = 1cos(β + α)− a cos(β − α) + b sin(β − α)×[(eiαh(z)z− e−iα g(z)z)− i(sin(β + α)− b cos(β − α)− a sin(β − α))] (9)is the Carathe´dory function under the condition A = cos(β+α)−a cos(β−α) + b sin(β − α) > 0. Then we have1− r1 + r≤ |p1(z)| ≤ 1 + r1− r (10)for p1(z) ∈ P and |z| = r < 1. If we substitute (9) into (10) and after simplecalculations we get(A+ |B|)− (A− |B|)r1 + r≤ |p(z)| ≤ (A+ |B|) + (A− |B|)r1− r . (11)Using (8) and (11) we obtain(A+ |B|)r − (A− |B|)r2(1 + r)|eiβ − e−iβz2| ≤|eiαh(z)− e−iαg(z)|≤ (A+ |B|)r + (A− |B|)r2(1− r)|eiβ − e−iβz2| .(12)On the other hand, we have11 + r2≤ 1|eiβ − e−iβz2| ≤11− r2 . (13)Therefore, if we use (13) in (12) we obtain the desired result.Theorem 2.2. Let f = h+ g be element of S∗H, then we have|h′(z)− e−2iαg′(z)| ≤ (A+ |B|) + (A− |B|)r(1− r)3for |z| = r < 1 where A = cos(β + α) − a cos(β − α) + b sin(β − α) > 0,B = sin(β + α) − b cos(β − α) − a sin(β − α), g′(0) = b1 = a + ib for somechoice of angles α and β.550 E. Yavuz DumanP r o o f. Using Lemma 2.1, we obtain thatzp′(z)p(z)= zAp′1(z)Ap1(z) + iBfor all z in the open unit disc. Also, we know that the following inequalitysatisfies for functions which in the Carathe´dory class:∣∣∣∣z p′(z)p(z)∣∣∣∣ =∣∣∣∣∣z p′1(z)p1(z) + iBA∣∣∣∣∣ ≤ 2r1− r2 . (14)On the other hand, from the equation (8) we havezh′(z)− e−2iαzg′(z)h(z)− e−2iαg(z) =1 + e−2iβz21− e−2iβz2 + zp′(z)p(z). (15)Considering (14) and (15) together, we obtain∣∣∣∣zh′(z)− e−2iαzg′(z)h(z)− e−2iαg(z)∣∣∣∣ = ∣∣∣∣1 + e−2iβz21− e−2iβz2 + z p′(z)p(z)∣∣∣∣≤∣∣∣∣1 + e−2iβz21− e−2iβz2∣∣∣∣+ ∣∣∣∣z p′(z)p(z)∣∣∣∣ . (16)Also we know that1− r21 + r2≤∣∣∣∣1 + e−2iβz21− e−i2β∣∣∣∣ ≤ 1 + r21− r2 . (17)Using (17) and (14) in (16), we get∣∣∣∣zh′(z)− e−2iαzg′(z)h(z)− e−2iαg(z)∣∣∣∣ ≤ ∣∣∣∣1 + e−2iβz21− e−2iβz2∣∣∣∣+ ∣∣∣∣z p′(z)p(z)∣∣∣∣ = 1 + r1− r . (18)Using Lemma 2.1 in (18), we obtain that|h′(z)− e−2iαg′(z)| ≤ (A+ |B|) + (A− |B|)r(1− r)3 .Lemma 2.3. Let ω(z) be the analytic dilatation of f = h + g ∈ SHdefined by ω(z) = g′(z)/h′(z) for all z ∈ D, then we have(1− r)(1− |b1|)1 + |b1|r ≤ |1− e−2iαω(z)| ≤ (1 + r)(1 + |b1|)1 + |b1|r (19)(|z| = r < 1) where g′(0) = b1 6= 0 and |b1| < 1.P r o o f. Let we define the functionφ(z) =ω(z)− b11− b1ω(z)SOME DISTORTION THEOREMS . . . 551where g′(0) = b1 for all z ∈ D. Since φ(z) is a transformation which mapsD onto itself we have |φ(z)| < 1 and φ(0) = 1. Thus we can writeω(z) ≺ b1 + z1 + b1z.On the other hand, the function ω(z) = b1+z1+b1zmaps |z| = r into the circlecentered atC(r) ={Reb1(1− r2)1− |b1|2r2 ,Imb1(1− r2)1− |b1|2r2},having the radiusρ(r) =(1− |b1|2)r1− |b1|2r2 .So we have ∣∣∣∣ω(z)− b1(1− r2)1− |b1|2r2∣∣∣∣ ≤ (1− |b1|2)r1− |b1|2r2 .Therefore, we get the result after some simple calculations.Theorem 2.4. Let f = h+ g be an element of S∗H, then we have|h′(z)| ≤ (1 + |b1|r)(A+ |B|+ (A− |B|r))(1− |b1|r)(1− r)4 , (20)|g′(z)| ≤ (A+ |B|) + (A− |B|)r(1− r)3(1 +1 + |b1|r(1− r)(1− |b1|r))(21)for |z| = r < 1 where A = cos(β + α) − a cos(β − α) + b sin(β − α) > 0,B = sin(β + α) − b cos(β − α) − a sin(β − α), g′(0) = b1 = a + ib for somechoice of angles α and β.P r o o f. Let consider the analytic dilatation function ω = g′/h′ off = h+ g. Then, we have|h′(z)− e−2iαg′(z)| = |h′(z)− e−2iαω(z)h′(z)|= |h′(z)||1− e−2iα(z)|. (22)Considering (19) and Theorem 2.2 in (22) we obtain,|h′(z)| ≤ (1 + |b1|r)(A+ |B|+ (A− |B|r))(1− |b1|r)(1− r)4 ,and|g′(z)| ≤ (A+ |B|) + (A− |B|)r(1− r)3(1 +1 + |b1|r(1− r)(1− |b1|r))for all |z| = r < 1.552 E. Yavuz DumanCorollary 2.5. Let f = h+ g ∈ S∗H, then we have|f(z)| ≤∫ r0(1 + |b1|ρ)((A+ |B|) + (A− |B|)ρ)(1− |b1|ρ)(1− ρ)4 dρ+∫ r0(A+ |B|) + (A− |B|)ρ(1− ρ)3(1 +1 + |b1|ρ(1− ρ)(1− |b1|ρ))dρfor |z| = r < 1 where A = cos(β + α) − a cos(β − α) + b sin(β − α) > 0,B = sin(β + α) − b cos(β − α) − a sin(β − α), g′(0) = b1 = a + ib for somechoice of angles α and β.P r o o f. For f = h+ g, we have the following inequalitiesf = h+ g =∫ r0h′(ρeiθ)eiθdρ+∫ r0g′(ρeiθ)eiθdρ=∫ r0h′(ρeiθ)eiθdρ+∫ r0g′(ρeiθ)e−iθdρ =∫ r0fz(ρeiθ)eiθdρ+∫ r0fz(ρeiθ)e−iθdρ.Hence|f | = |h+ g| ≤ |h|+ |g| ≤∫ r0|fz(ρeiθ)|dρ+∫ r0|fz(ρeiθ)|dρ⇒|f | ≤∫ r0|h′(ρeiθ)|dρ+∫ r0|g′(ρeiθ)|dρ⇒ .Applying inequalities (20) and (21) to the above, we obtain the result.References[1] J. Clunie and T. Sheil-Small, Harmonic univalent functions. Ann. Acad.Aci. Fenn. Ser. A. I. Math. 9 (1984), 3-25.[2] P. Duren, Harmonic Mappings in the Plane. Cambridge University Press,New York, 2004.[3] A.W. Goodman, Univalent Functions, Volume 1. Mariner PublishingCompany, Inc., Florida, 1983.Department of Mathematics and Computer ScienceFaculty of Scinece and LettersI˙stanbul Ku¨ltu¨r University,I˙stanbul – 34156, TURKEYe-mail: e.yavuz@iku.edu.tr Received: GFTA, August 27-31, 2010

Some Distortion Theorems for Starlike Harmonic Functions

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Some Distortion Theorems for Starlike Harmonic Functions

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