Radii of harmonic mapping with fixed second coefficients in the plane

DOI: 10.24193/subbmath.2018.2.03

Radii of harmonic mapping with fixed second coefficients in the plane

Rasoul Aghalary and Ali Mohammadian

Abstract. In this paper we investigate the radii problem for harmonic functions with a fixed coefficient and determine the radii of univalence, stable starlikness, stable convexity, fully starlikness and fully convexity of orderαfor these type of functions. All results are sharp. Also these results generalize and improve some results in the literature.

Mathematics Subject Classification (2010):30C45, 30C80.

Keywords:Stable starlike functions, stable univalent function, stable convex func- tion, radii problem.

1. Introduction and Preliminaries

Denote by Hthe class of all complex-valued harmonic functions f in the unit disk D={z ∈C:|z|<1} normalized byf(0) = 0 =fz(0)−1. Each f ∈ H can be decomposed asf =h+g, where gandhare analytic inDso that

h(z) =z+

∞

X

n=2

anzⁿ and g(z) =

∞

X

n=1

bnzⁿ. (1.1) Let SH denote the class of univalent and orientation- preserving functions f ∈ H.

Then the Jacobian off is given byJ_f(z) =|h⁰(z)|²− |g⁰(z)|².

A necessary and sufficient condition (see Lewy [11]) for a harmonic functionf to be locally univalent inDis that Jf(z)>0 inD.

Let KH,S_H^∗ and CH be the subclass of SH mapping D onto convex, starlike and close-to-convex domains, respectively. Also denote byK⁰_H,S^∗0_H,C_H⁰ andS_H⁰ the class consisting of the functions f in KH,S_H^∗,CH and SH respectively, for whichfz(0) = b₁= 0.

One of the important questions in the study of classS_H⁰ and its subclasses is related to coefficient bounds. In 1984, Clunie and Sheil-Small [5], it was conjectured that

the Taylor coefficients of the the series hand g, namely, an and bn are satisfy the conditions|an| ≤An and|bn| ≤Bn, where

An= 1

6(2n+ 1)(n+ 1) and Bn=1

6(2n−1)(n−1). (1.2) The harmonic functionf(z) =z+

∞

X

n=2

A_nzⁿ+

∞

X

n=2

B_nzⁿ is known as harmonic Koebe function and maps the unit disk Donto the slit plane C\ {u+iv:u≤ −1/6, v= 0}

which is starlike function inD.

Although, the coefficients conjecture remains an open problem for the full classS_H⁰, the same has been verified for all functions f ∈ S_H⁰ with real coefficients and all function f ∈ S_H⁰ for which either f(D) is starlike with respect to the origin, close- to-convex function and for convex in one direction (see [5], [19], [20]). The extremal function is the harmonic Koebe function. Iff ∈ K_H⁰,Clunie and Sheil-Small [5] proved the Taylor coefficients ofhandg satisfy the inequality

|an| ≤ n+ 1

2 and |bn| ≤ n−1

2 , (1.3)

and equality occurs for the harmonic half-plane mapping L(z) = 2z−z²

2(1−z)² + −z² 2(1−z)²

=

∞

X

n=1

n+ 1 2 zⁿ−

∞

X

n=1

n−1 2 zⁿ.

Chaqui et al. [4] introduced the notion of fully starlike and fully convex harmonic function that do inherit the properties of starlikeness and convexity, respectively. A harmonic mappingf ofDis said to be fully convex of orderα, 0≤α <1, if it maps every circle|z|=r <1 in a one-to-one manner onto a convex curve satisfying

∂

∂θ(arg ∂

∂θf(reⁱθ))

> α, 0≤θ < tπ, 0< r <1.

Similarly, a harmonic mappingf ofDis said to be fully starlike of orderα, 0≤α <1, if it maps every circle|z|=r <1 in a one-to-one manner onto a curve that bound a domain starlike with respect to the origin satisfying

∂

∂θ(arg(f(reⁱθ)))> α, 0≤θ < tπ, 0< r <1.

In [5] Clunie and Sheil-Small proved the following result

Lemma 1.1.Ifh, g are analytic inDwith|h⁰(0)|>|g⁰(0)|andh+gis close-to-convex for each ,||= 1, thenf =h+g is close-to-convex inD.

This lemma has been used to obtain many important results. Motivated by this result Hernandes et al. [7] introduced the notion of stable univalent stable starlike, stable convex and stable close-to-convex harmonic functions.

We say that the (sense-preserving) harmonic mappingf =h+g is stable univalent (resp. stable convex, stable starlike with respect to origin, or stable close-to-convex)

if all the mappingfλ=h+λgwith|λ|= 1 are univalent (resp. convex, starlike with respect to origin, or close-to-convex) inD.

Analogs to Lemma 1.1 it is proved [7] that the harmonic mappingf =h+gis stable univalent (resp. stable convex, stable starlike with respect to origin, or stable close- to-convex) if and only if all the mappingfλ=h+λgwith|λ|= 1 are univalent (resp.

convex, starlike with respect to origin, or close-to-convex) inD.

We note that fully starlike function need not be univalent, but stable starlike function is univalent. Also it is easy to see that the stable starlike and stable convex functions are fully starlike and fully convex functions, respectively. It is easy to see that a functionf =h+g whose coefficients satisfy the conditions|an| ≤An and|bn| ≤Bn

is harmonic inD, however it need not be univalent. For example the functionf(z) = z+ 2z⁴ satisfy the above mentioned conditions but it is not sense-preserving in D. It is therefore of interest to determine the largest subdisk |z| < ρ <1 in which the functionf satisfying the condition (1.2) and (1.3) (or others) influence the univalency off. Recall that given two subsetsMandN ofHtheN radius in Mis the largest Rsuch that, for everyf ∈ M,r⁻¹f(rz)∈ N for everyr≤R.

The radius of fully convexity of the class K_H⁰ is √

2−1, while the radius of fully convexity of the classS^∗0_His 3−√

8 (see [18], [19]). The corresponding problem for the radius of fully starlikeness are still unsolved. In [10], the radius of close-to-convexity of harmonic mapping was determined. These results are generalized in context of fully starlike and fully convex functions of orderα(0≤α <1) in [14]. Also, we remark that many authors studied the radii problem in the analytic functions (see [1], [2], [11], [13], [15], [17], [16], [21]) There is a challenge in fixing the second coefficient which is due to that removal of natural extremal function from the class. In this paper such as [12] we investigate the radii of univalence, stable starlikeness, stable convexity, fully starlikeness and fully convexity for these types of functions. Also we determine the Bloch constant for harmonic functions with fixed second coefficient.

For proving our result we shall need the following result due to Jahangiri [8], [9].

Theorem 1.1. Let f =h+g, where hand g are given by (1.1) and let 0 ≤α < 1.

Then we have the following (i) If

∞

X

n=2

n−α 1−α|a_n|+

∞

X

n=1

n+α

1−α|b_n| ≤1, thenf is harmonic univalent andf is fully starlike of orderα.

(ii) If

∞

X

n=2

n(n−α) 1−α |an|+

∞

X

n=1

n(n+α)

1−α |bn| ≤1, thenf is harmonic univalent andf is fully convex of order α.

By making use of Theorem 1.1 we conclude the following result.

Corollary 1.2. Let f = h+g, where h and g are given by (1.1), then we have the following

(i) If

∞

X

n=1

n(|an|+|bn|)≤2 with a1= 1, thenf is harmonic univalent andf is stable starlike.

(ii) If

∞

X

n=1

n²(|an|+|bn|)≤2 with a1= 1, thenf is harmonic univalent andf is stable convex.

2. Radii of Univalence

In this section, the sharp stable starlike (convex) radius and the sharp fully starlike (convex) of orderα,(0≤α <1) are obtained for harmonic functions.

Theorem 2.1. Letf =h+g∈ H,where h, g is given by(1.1). Let

b1= 0, |a2|+|b2|= 2b (0≤b≤1) and |an|+|bn| ≤n(n≥3). (2.1) Then for f,

(i) the radius of stable starlikeness isrs, wherers=rs(b)is the smallest root in(0,1) of the equation

1 +r= 2[1 + 2(1−b)r](1−r)³. (2.2) (ii) the radius of stable convexity isrc, whererc =rc(b)is the smallest root in (0,1) of the equation

2[1 + 4(1−b)r](1−r)⁴= 1 + 4r+r². (2.3) Furthermore, all results are sharp.

Proof. First we prove the case (i). Forr∈(0,1) withr≤r_s,it is sufficient to show thatF_r is stable starlike, where

F_r(z) =z+

∞

X

n=2

a_nrⁿ⁻¹zⁿ+

∞

X

n=2

b_nrⁿ⁻¹zⁿ. (2.4) According to Corollary 1.1, it is sufficient to show that

∞

X

n=2

n(|an|+|bn|)rⁿ⁻¹≤1. (2.5) Considering condition (2.1), we have

∞

X

n=2

n(|an|+|bn|)rⁿ⁻¹≤4brs+

∞

X

n=3

n²rⁿ⁻¹_s

= 4br_s+ 1 +r_s

(1−rs)³−1−4r_s= 1,

provided rs is the root of the equation (2.2) in the hypothesis of the theorem. Set m(r) = 1 +r−2(1 + 2(1−b)r)(1−r)³.

Then m(0) = −1 and m(1) = 2 and so intermediate value theorem shows that the equation (2.2) has a root in the interval (0,1). Next for proving the sharpness part we consider the functionf0=h0+g0,where

h₀(z) = 2z+3

2z²− 2z−z²

2(1−z)² and g₀(z) = 2bz²−1

2z²+ z² 2(1−z)². Direct calculation leads to

h⁰₀(z) = 2 + 3z− 1

(1−z)³ and g⁰₀(z) = 4bz−z+ z (1−z)³. Now from (2.2), we have

[h⁰₀(r)−g₀⁰(r)]_r=rs= 1

(1−r)³[2(1 + 2r(1−b))(1−r)³−(1 +r)]|r=rs = 0.

Hence,

J_f0(r)|r=rs = [(h⁰₀(r) +g₀⁰(r))(h⁰₀(r)−g⁰₀(r))]_r=rs= 0.

Therefore, in view of Lewy’s theorem, the function f0 is not univalent in|z| < r if r > rs. This shows the radiusrsis sharp.

Case (ii). The proof of (2.3) is similar to the case (i) and we omit the details. To proof sharpness, we take the functionf1=h1+g1, where

h1(z) = 2z+ 2z²− z

(1−z)² and g1(z) =−2bz² Then if we defineF₁(z) =h₁(z) +g₁(z), it yields

Re

1 + zF₁⁰⁰(z) F₁⁰(z)

|r=r_c=

2(1−r⁴)[1 + 4(1−b)r]−(1 + 4r+r²) (1−r)[(2 + 4(1−b)r)(1−r³)−(1 +r)]

|r=r_c= 0.

The denominator of the rational function in the middle of the equation above is greater than the numerator for all 0≤r <1 and 0≤b≤1. Therefore, if we take the smallest root of the denominator with r_p in (0,1) then, we have r_c < r_p. Hence for r_c < r < r_p the denominator of the rational function in the middle of the equation above is the positive while the numerator is negative, and this means the expression in the middle is negative for r_c < r < r_p. So we observe that the functionF₁ is not convex for|z|< r, wherer > r_c. Now the functionsF_λ=h₁(z) +λg₁(z) for allλwith

|λ|= 1 are not convex for|z|< r, wherer > rc, or the functionsFλ=h1(z) +λg1(z) for all λwith |λ|= 1 are not convex for|z|< r, wherer > rc. This shows that the

radiusr_c is sharp.

For example let us consider the half-plane harmonic functionL=h+g, where h(z) = 2z−z²

2(1−z)², g(z) = −z² 2(1−z)².

This function mapsD harmonically onto domain Ω ={z ∈C: Re{z}>−1/2} and so it is convex function. We remark this function is not stable starlike function. Since if we consider the function

h(z) = 2z−z² 2(1−z)²

we obtain

Rezh⁰(z) h(z) = 2Re

1

1−z− 1 2−z

,

and with calculation one can see that which is zero in the pointz₀ =

√14

4 e^{icos−1√314}. Hence h is not starlike and by result in [7] the half plane mapLis not stable starlike.

On the other hand it is easy to see that L(z) =z+

∞

X

n=2

n+ 1 2 zⁿ−

∞

X

n=2

n−1 2 zⁿ,

so by takinga₂= 3/2, b₂= 1/2 we observe thatL satisfies the condition of Theorem 2.1. ThereforeLis stable starlike function in the disk|z| ≤r₀wherer₀'0.164878 is the real root of the equation 2r³−6r²+ 7r−1 = 0, in the interval (0,1).

Theorem 2.2. Letf =h+g∈ H,where h, g is given by(1.1). Let

|b₁|<1, |a₂|+|b₂|= 2b(0≤b≤M

2 ) and |a_n|+|b_n| ≤M (M >0) (n≥3). (2.6) Then for f,

(i) the radius of stable starlikeness isrs, wherers=rs(b)is the smallest root in(0,1) of the equation

M−[1 +M− |b1|+ 2(M −2b)r](1−r)²= 0. (2.7) (ii) the radius of stable convexity isrc, whererc =rc(b)is the smallest root in (0,1) of the equation

(1−r)³[1 +M − |b1|+ 4r(M−2b)] =M(1 +r). (2.8) Furthermore, all results are sharp.

Proof. The proof of case (i) is similar to the proof of case (i) in the Theorem 2.1 and so is omitted. The functionf₀=h₀+g₀,where

h₀(z) =z− M z³

2(1−z) and g₀(z) =−|b1|z−2bz²− M z³

2(1−z) (2.9) shows that the result is sharp. Indeed, in view of (2.9) and direct computation imply that

h⁰₀(z) = 1−M3z²−2z³

2(1−z)² and g₀⁰(z) =−|b1| −4bz−M3z²−2z³ 2(1−z)², and so

Jf₀(r) =|h⁰₀(r)|²− |g⁰₀(r)|²= (1 + 4br)

1−4br− |b1| −M r²

3−2r (1−r)²

, which shows thatJf₀(rs) = 0 andJf₀(r)<0 for r > rs. Thus the proof of case (i) is complete.

Case (ii). For r∈(0,1) withr≤rc,it is sufficient to show thatFr is stable convex, where

Fr(z) =z+

∞

X

n=2

anrⁿ⁻¹zⁿ+

∞

X

n=1

bnrⁿ⁻¹zⁿ. (2.10)

In view of corollary 1.1, it is sufficient to show that

|b₁|+

∞

X

n=2

n²(|a_n|+|b_n|)rⁿ⁻¹≤1. (2.11) Considering condition (2.6), we have

|b1|+

∞

X

n=2

n²(|an|+|bn|)rⁿ⁻¹≤ |b1|+ 8brc+M

∞

X

n=3

n²r_cⁿ⁻¹

=|b1|+ 8brc+M

1 +rc

(1−rc)³ −1−4rc

= 1, provided rc is the root of the Equation (2.8) in the hypothesis of the theorem. Set

m(r) =M[1 +r]−[1 +M− |b1|+ 4(M −2b)r](1−r)³.

Then m(0) =−(1− |b1|) and m(1) = 2M and so intermediate value theorem shows that the Equation (2.8) has a root in the interval (0,1). To proof of sharpness, we define the functionf1=h1+g1,where

h1(z) =z−M z³

2(1−z) and g1(z) =−|b1|z−2bz²−M z³ 2(1−z). Then if we consider the function F1(z) =h1(z) +g1(z), direct calculation gives that F₁⁰(r_c) +r_cF₁⁰⁰(r_c) = 0. In the other words we conclude that

Re

1 +zF₁⁰⁰(z) F₁⁰(z)

|_z=rc

= 0.

The rest of proof is exactly the same as proof of sharpness part of case (ii) of Theorem

2.1 and we omit the details.

There are two important constants, one is the radius of univalencs, while the other is the Bloch constant. Many authors have been studied these problems. (see [3]). By making use of Theorem 2.2 we conclude the following result.

Corollary 2.3. Let f =h+g∈ H, whereh, g is given by (1.1). Let b1= 0, |a2|+|b2|= 2b, 0≤b≤2c

π and |f(z)|< c. (2.12) Then for f, the radius of univalence is r0, where r0 is the smallest root in (0,1) of the equation

4c π −

1 + 4c

π + 4 2c

π −b

r

(1−r)²= 0. (2.13) Furthermore,f(Dr₀)contains a univalent disk of radius at least

R0=r0−2br02

− 4cr₀³

π(1−r0), (2.14)

whereDr0={z∈C:|z|< r₀}.

Proof. According to [3] we can obtain the sharp estimates

|an|+|bn| ≤ 4c π

for anyn≥3. By Theorem 2.2 withM =^4c_π,we conclude that the radius of univalence isr₀. Furthermore, for|z|=r₀, we have

|f(z)|=|z+

∞

X

n=2

(anzⁿ+bnzⁿ)|

≥ |z| − |

∞

X

n=2

(a_nzⁿ+b_nzⁿ)|

≥r0−

∞

X

n=2

(|an|+|bn|)rⁿ₀

≥r0−2br₀²−4c π

∞

X

n=3

rⁿ₀

=r0−2br₀²− 4cr₀³

π(1−r0)=R0

and the proof is complete.

Theorem 2.4. Letf =h+g∈ H,where h, g is given by(1.1). Let b1= 0, |a2|=a,|b2|=b, |an| ≤ n+ 1

2 and |bn| ≤ n−1

2 f or n≥3, (2.15) where0≤a≤³₂ and0≤b≤ ¹₂.Then for f,

(i) the radius of fully starlikeness is rs, where rs = rs(α, b) is the smallest root in (0,1) of the equation

(1 +r)−α(1−r)²= (1−r)³{r[−2(a+b) +α(a−b) + 4−α] + 2(1−α)}. (2.16) (ii) the radius of fully convexity isrc, where rc =rc(α, b)is the smallest root in(0,1) of the equation

1 + 4r+r²−α(1−r)²= (1−r)⁴{r[(−4(a+b) + 2α(a−b) + 8−2α)] + 2(1−α)}. (2.17) Furthermore, all results are sharp.

Proof. The proof is similar to the proof of Theorem 2.1. Letr∈(0,1), it is sufficient to prove thatFr(z) is fully starlike of orderα, whereFr(z) is given by (2.10). According to Theorem 1.1 and assumption (2.15), it is enough to show that

(2−α)ar+ (2 +α)br+

∞

X

n=3

[n²−α]rⁿ⁻¹≤1−α.

By making use of identities,

∞

X

n=1

n²rⁿ⁻¹= 1 +r (1−r)³ and

∞

X

n=1

rⁿ⁻¹= 1 (1−r),

the last inequality reduces to

(1 +r)−α(1−r)²−(1−r)³[r(−2(a+b) +α(a−b) + 4−α) + 2(1−α)]≥0.

Set

m(r) = (1 +r)−α(1−r)²−(1−r)³[r(−2(a+b) +α(a−b) + 4−α) + 2(1−α)].

Thenm(0) =−(1−α) andm(1) = 2 and so intermediate value theorem shows that the Equation (2.16) has a root in the interval (0,1). Therefore,Fr(z) is fully starlike of order αforr≤rs, wherersis smallest root of equation (2.16) in (0,1). To prove sharpness, we takef₀=h₀+g₀, where

h0(z) = 2z−az²+3

2z²− 2z−z²

2(1−z)² and g0(z) =bz²−1

2z²+ z² 2(1−z)². Direct computation leads to

h⁰₀(z) = 2−2az+ 3z− 1

(1−z)³ , g₀⁰(z) = 2bz−z+ z (1−z)³, and

∂

∂θ arg f0(re^iθ)

|θ=0= rh⁰₀(r)−rg⁰₀(r) h₀(r) +g₀(r) =

2−2ar+ 4r−2br−_(1−r)^1+r3

2−ar+r+br−_1−r¹ . (2.18) Also, from equation (2.16) we have

[2−2ar+ 4r−2br](1−r)³−(1 +r)

[2−ar+r+br](1−r)³−(1−r)² =α. (2.19) Thus in view of (2.18) and (2.19) we conclude that

∂

∂θ arg f0(re^iθ)

|θ=0,r=r_s =α,

and this shows that the bound r_s is best possible. The proof of first part of case (ii) is the same as the case (i) and we omit the details. For the sharpness, we take f₁=h₁+g₁, where

h₁(z) = 2z−az²+3

2z²− 2z−z²

2(1−z)² and g₁(z) =−bz²+1

2z²− z² 2(1−z)². Direct computation leads to

h⁰₁(z) = 2−2az+ 3z− 1

(1−z)³ , g₁⁰(z) =−2bz+z− z (1−z)³, h⁰⁰₁(z) =−2a+ 3− 3

(1−z)⁴ , g₁⁰⁰(z) =−2b+ 1− 1 + 2z (1−z)⁴,

and

∂

∂θ

arg ∂

∂θf1(re^iθ)

|θ=0 (2.20)

= h⁰₁(r) +g₁⁰(r) +r(h⁰⁰₁(r) +g₁⁰⁰(r)) h⁰₁(r)−g₁⁰(r)

= 2−4ar+ 8r−4br−^1+4r+r_(1−r)4²

2−2ar+ 2r+ 2br−_(1−r)¹ 2

. Also, from equation (2.18) we have

[2−4ar+ 8r−4br](1−r)⁴−(1 + 4r+r²)

[2−2ar+ 2r+ 2br](1−r)⁴−(1−r)² =α. (2.21) Thus in view of (2.20) and (2.21) we conclude that

∂

∂θ(arg(∂

∂θf1(re^iθ)))|θ=0,r=rc =α,

and this shows that the boundrc is best possible. Hence the proof is complete.

Remark 2.5. By compering Theorems 2.1 and 2.3 one can observe that by putting α= 0 anda+b= 2con the Theorem 2.3 we obtain the assumption of Theorem 2.1.

Also, by putting α = 0 on the equations (2.16) and (2.17) we obtain the equations (2.2) and (2.3), respectively. So in this case the radius of stable starlikeness and stable convexity is the the same as radius of fully starlikeness and fully convexity of order zero, respectively.

Corollary 2.6. Under the hypothesis of Theorem2.3 forf, we have

(i) the radius of stable starlikeness is rs, wherersis the smallest root in(0,1) of the equation

(1 +r) = (1−r)³[r(−2(a+b) + 4) + 2]. (2.22) (ii) the radius of fully convexity is rc, where rc is the smallest root in (0,1) of the equation

1 + 4r+r²= (1−r)⁴{r[(−4(a+b) + 8] + 2}. (2.23) Furthermore, all results are sharp.

Proof. The proof is similar to the proof of Theorem 2.3 and we omit the details.

Theorem 2.7. Let f =h+g∈ H is given by (1.1) and|an| ≤An and|bn| ≤Bn for n ≥ 3, where An and Bn is given by (1.2). Also, let b1 = 0,|a2| = a and |b2| =b, where0≤a≤⁵₂ and0≤b≤ ¹₂.Then for f,

(i) the radius of fully starlikeness isrs, wherers=rs(b)is the smallest root in (0,1) of the equation

(1 +r)²−α(1−r)²= (1−r)⁴{r[−2(a+b) +α(a−b) + 2(3−α)] + 2(1−α)}, (2.24) (ii) the radius of fully convexity is rc, where rc =rc(b) is the smallest root in (0,1) of the equation

(1+r)[1+6r+r²−α(1−r)²] = (1−r)⁵{r[(−4(a+b)+2α(a−b)+12−4α)]+2(1−α)}, (2.25)

Furthermore, all results are sharp.

Proof. Using the same argument of the proof of Theorem 2.3 we can obtain the equations (2.24) and (2.25). Only we prove the sharpness part of theorem. In the first case the results is sharp for the functionf0=h0+g0given by

h0(z) = 2z−az²+5

2z²−z−¹₂z²+¹₆z³

(1−z)³ and g0(z) =bz²−1 2z²+

1

2z²+¹₆z³

(1−z)³ . (2.26) In view of (2.26), and computation shows that

∂

∂θ(arg(f₀(re^iθ)))|_θ=0= rh⁰₀(r)−rg⁰₀(r)

h₀(r) +g₀(r) = 2−2ar+ 6r−2br−^(1+r)_(1−r)²4

2−ar+ 2r+br−_(1−r)¹ ₂ . (2.27) Also, from equation (2.24) we have

[2−2(a+b)r+ 6r](1−r)⁴−(1 +r)²

[2−(a−b)r+ 2r](1−r)⁴−(1−r)² =α. (2.28) Thus in view of (2.27) and (2.28) we conclude that

∂

∂θ(arg(f₀(re^iθ)))|_θ=0,r=rs =α,

and this shows that the bound rs is best possible. Furthermore, in the second part the result is sharp for the functionf1=h1+g1 given by

h1(z) = 2z−az²+5

2z²−z−¹₂z²+¹₆z³

(1−z)³ and g1(z) =−bz²+1 2z²−

1

2z²+¹₆z³ (1−z)³ .

(2.29) According to (2.29), direct calculation yields

∂

∂θ

arg ∂

∂θf1(re^iθ)

|θ=0 (2.30)

= h⁰₁(r) +g₁⁰(r) +r(h⁰⁰₁(r) +g₁⁰⁰(r)) h⁰₁(r)−g₁⁰(r)

= [2−4ar+ 12r−4br](1−r)⁵−(1 +r)²(1−r)−6r−8r²−2r³ [2−2ar+ 4r+ 2br](1−r)⁵−(1 +r)(1−r)² . Meanwhile, from equation (2.25) we have

[2−4ar+ 12r−4br](1−r)⁵−(1 +r)²(1−r)−6r−8r²−2r³

[2−2ar+ 4r+ 2br](1−r)⁵−(1 +r)(1−r)² =α (2.31) Thus in view of (2.27) and (2.28) we conclude that

∂

∂θ

arg ∂

∂θf₁(re^iθ)

|θ=0,r=rc=α,

and this shows that the boundr_c is best possible. Hence the proof is complete.

Acknowledgement. The authors would like to thank the referee for many useful sug- gestions.

References

[1] Ali, R.M., Cho, N.E., Jain, N.K., Ravichandran, V.,Radii of starlikeness and convexity of functions defined by subordination with fixed second coefficients, Filomat,26(2012), 553-561.

[2] Ali, R.M., Nagpal, S., Ravichandran, V., Second-order differential subordinations for analytic functions with fixed initial coefficient, Bull. Malays. Math. Sci. Soc.,34(2011), no. 2, 611-629.

[3] Chen, Sh., Ponnusamy, S., Wang, X.,Bloch constants and Landau’s theorems for planner p-harmonic mapping, J. Math. Anal. Appl.,373(2011), 102-110.

[4] Chuaqui, M., Duren, P.L., Osgood, B.,Curvature properties of planar harmonic map- pings, Comput. Methods Funct. Theory,4(2004), no. 1, 127-142.

[5] Clunie, J.G., Sheil-Small, T.,Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser.

A.,9(1984), 3-25.

[6] Gangadharan, A., Ravichandran, V., Shanmugam, T.N.,Radii of convexity and strong starlikeness for some classes of analytic functions, Journal of Mathematical Analysis and Applications,211(1997), no. 1, 301-313.

[7] Hernandez, R., Martin, M.J.,Stable geometric properties of analytic and harmonic func- tions, Math. Proc. of the Cambridge Philosophical Soc.,155(2013), no. 2, 343-359.

[8] Jahangiri, J.M.,Coefficient bounds and univalence criteria for harmonic functions with negative coefficients, Ann. Univ. Mariae Curie-Shlodowska, sect. A, 59(1998), no. 2, 57-66.

[9] Jahangiri, J.M., Harmonic functions starlike in the unit disk, J. Math. Anal. Appl., 235(1999), no. 2, 470-477.

[10] Kalaj, D., Ponnusamy, S., Vuorinen, M.,Radius of close-to-convexity of harmonic func- tions, Complex Var. Elliptic Equ,59(2014), no. 4, 539-552.

[11] Lewy, H.,On the non-vanishing of the Jacobian in certain one-to-one mapping, Bull.

Amer. Math. Soc.,24(10)(1936), 689-692.

[12] Long, B.Y., Hang, H.,Radii of harmonic mapping in the plane, J. Aust. Math. Soc., (2016), 1-7.

[13] Ma, W., Minda, D.,Uniformly convex functions, Ann. Polon. Math.,57(1992), 165-175.

[14] Nagpal, S., Ravichandran, V.,Fully starlike and convex harmonic mappings of orderα, Ann. Polon. Math.,108(2012), 1-7.

[15] Rajni, M., Nagpal, S., Ravichandran, V., Radii of starlikeness and convexity for an- alytic functions with fixed second coefficient satisfying certain coefficient inequalities, Kyungpook Mathematical Journal,55(2015), no. 2, 395-410.

[16] Ravichandran, V., Radii of starlikeness and convexity of analytic functions satisfying certain coefficient inequalities, Mathematica Slovaca,64(2014), no. 1, 27-38.

[17] Ravichandran, V., Khan, M.H., Silverman, H., Subramanian, K.G.,Radius problems for a class of analytic functions, Demonstratio Math.,39(2006), no. 1, 67-74.

[18] Ruscheweyh, St., Salinas, L.,On the preservation of direction-convexity and Goodman- Saff Conjecture, Ann. Acad. Sci. Fenn. Ser. A I Math.,14(1989), 63-73.

[19] Sheil-Small, T., Constants for plannar harmonic mapping, J. Lond. Math. Soc., 42(1990), no. 2, 237-248.

[20] Wang, X.-T., Liang, X.-Q., Zhang, Y.-L.,Precise coefficient for close-to-convex harmonic univalent mapping, J. Math. Anal. Appl.,263(2001), no. 2, 501-509.

[21] Yamashita, S.,Radii of univalence, starlikeness and convexity, Bull. Austral. Math. Soc., 25(1982), 453-457.

Rasoul Aghalary

Urmia University, Faculty of Science, Department of Mathematics Urmia, Iran

e-mail:[email protected] and [email protected] Ali Mohammadian

Urmia University, Faculty of Science, Department of Mathematics Urmia, Iran

e-mail:[email protected]