NEW CLASSES OF HARMONIC FUNCTIONS DEFINED BY FRACTIONAL OPERATOR

(1)

F.M ¨UGE SAKAR1_{, YASEMIN BA ˘}_GCI2_{, H. ˝}_{OZLEM G ¨}_UNEY3_, _§

Abstract. In the present study, we introduce an investigation of new subclasses of har-monic functions which are defined by fractional operator. Firstly, using by fractional operator, we define new subclasses of harmonic functions. Later, we obtain main theo-rems of our study which contain sufficient and necessary coefficient bounds for functions related to the classes newly defined. Also, several particular characterization proper-ties of these classes are given. Some of these properproper-ties involve extreme points, convex combination, distortion bounds. Finally, several corollaries of the main theorems are presented.

Keywords: Harmonic functions, fractional operator, coefficient estimates, univalent.

AMS Subject Classification: 30C45, 30C50

˙

1. Introduction

Let u, v be real harmonic functions in the simply connected domain Ω, then the con-tinuous function f = u+iv defined in Ω is said to be harmonic in Ω. In any simply connected domain Ω ⊂ _C we can write f = h+g, where h and g are analytic in Ω. A necessary and sufficient condition forf to be locally univalent and sense-preserving in Ω is that |h0(z)| > |g0(z)| in Ω (see [5]).Let H be the family of functions f = h+g which are harmonic univalent and sense-preserving in the open unit disk U ={z:|z|<1}that f = h+g normalized by f(0) = h(0) = f0(0)−1 = 0. Let HW be the set harmonic univalent and sense-preserving functions inU [1] of the formf(z) =h(z) +g(z), where

h(z) =z+ ∞

X

n=2

anzn, g(z) = ∞

X

n=1

bnzn |b1|<1 (1)

are analytic inU.

1

Batman University, Faculty of Management and Economics, Department of Business Administration, Batman-Turkey, e-mail: mugesakar@hotmail.com; ORCID: http://orcid.org/ 0000-0002-3884-3957.

2 _{Batman University, Institute of Science, Master’s Degree in Mathematics, Batman, Turkey.}

e-mail: yasemin−bagci72@hotmail.com; ORCID: http://orcid.org/ 0000-0003-1083-4829.

3 _{Dicle University, Faculty of Science, Department of Mathematics, Diyarbakır-Turkey.}

e-mail: ozlemg@dicle.edu.tr; ORCID: http://orcid.org/ 0000-0002-3010-7795. § Manuscript received: March 21, 2017; accepted: April 20, 2017.

(2)

LetHW be the subclass ofHW consisting of functions of the form f(z) =h(z) +g(z), where

h(z) =z−

∞

X

n=2

|an|zn, g(z) = ∞

X

n=1

|bn|zn |b1|<1. (2)

In 1984 Clunie and Sheil-Small [5] investigated the class SH as well as its geometric subclasses and obtained some coefficient bounds. Since then, there has been several related papers on SH and its subclasses studied by Avcı and Zlotkiewicz [3], Silverman [11], Silverman and Silvia [13], Jahangiri [6] and Jahangiri et al. [7]. Furthermore, some other authors (e.g see [2] and [14]) have recently studied the harmonic univalent functions using by certain operators. In current study, a new subclasses of harmonic functions has been investigated by using fractional operator especially.

A fairly complate treatment, with applications of the fractional calculus, is given in the books [10] by Oldham and Spanier, and [9] by Miller and Ross. We refer to [12] for more insight into the concept of the fractional calculus. For further details on the materials in this paper see [8].

For convenience, we shall remind some definitions of the fractional calculus (i.e., frac-tional integral and fracfrac-tional derivative).

The fractional integral of order λfor analytical functionf(z) inD is defined by

D_z−λf(z) = 1 Γ(λ)

Z z

0

f(ζ)

(z−ζ)1−λdζ,(λ >0) (3) where the multiplicity of (z−ζ)(λ−1) is removed by requiringlog(z−ζ) to be real when (z−ζ)>0.

The fractional derivative of order λfor an analytic functionf(z) in D is defined by

Dλ_zf(z) = d dz(D

−λ

z f(z)) = 1 Γ(1−λ)

d dz

Z z

0

f(ζ)

(z−ζ)λdζ,(0≤λ <1), (4)

where the multiplicity of (z−ζ)−λ is removed by requiringlog(z−ζ) to be real when (z−ζ)>0.

Under the hypothesis of the fractional derivative, the fractional derivative of order (n+λ) for an analytic functionf(z) in Dis defined by

D_zn+λf(z) = d n

dzn(D λ

zf(z)), (0≤λ <1, n∈N0 ={0,1,2, ...}). (5)

From the definitions of the fractional calculus, we see that

D_z−λzk = Γ(k+ 1) Γ(k+ 1 +λ)z

k+λ_, _{(λ >}_{0, k >}₀₎ ₍₆₎

D_zλzk = Γ(k+ 1) Γ(k+ 1 +λ)z

k−λ

(3)

and

D_zn+λzk= Γ(k+ 1) Γ(k+ 1−λ−n)z

k−n−λ

, (0≤λ <1, k >0, n∈N0, k−n6= −1,−2, ...).

(8) Therefore, we see that for any real λ,

Dλ_zzk= Γ(k+ 1) Γ(k+ 1−λ)z

k−λ

, (k >0, k−λ6= −1,−2, ...). (9)

Using the rule of the fractional derivative, which is mentioned in the preceding, Aydo˘gan et al. [4] defined theλ−f ractional operator as follows,

f(z) =z+a2z2+...+anzn+...⇒Dzλf(z) =Dλz(z+a2z2+...+anzn+...)

D_zλf(z) = Γ(2−λ)zλD_zλf(z) =z+ ∞

X

n=2

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) anz

n_. ₍₁₀₎

From the definition ofDλf(z) we have the following properties:

i.D0f(z) =Df(z) =limλ→1Dλf(z) =zf0(z);

ii.Dλ(Dδf(z)) =Dδ(Dλf(z)) =z+ ∞

X

n=2

Γ(2−λ)Γ(2−δ)(Γ(n+ 1))2 Γ(n+ 1−λ)Γ(n+ 1−δ) anz

n_;

iii.D(Dδf(z)) =z+ ∞

X

n=2

nΓ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) anz

n₌_z(Dδ_f_(z))0

= Γ(2−λ)zλ(λDλ_z+zDλ_z+1f(z));

iv.D D λ_f_(z)

Dλ_f_(z) =z f0(z)

f(z), f or λ= 0

= 1 +zf 00_(z)

f0_(z), f or λ= 1.

Thus, we define the following class of functions.

Definition 1.1. Let f(z) = h(z) +g(z), be the harmonic univalent function given by Eq.(1), thenf ∈HW(λ, β, t, α) if and only if

<

(1−β)(1−t)D λ_f_(z)

z + (β+t)

(Dλf(z))0 z0 +βt

(Dλf(z))00 z00 −2βt

≥α (11)

where 0≤α <1, β ≥0, 0≤t≤1 and z=reiθ ∈U.

We also let HW(λ, β, t, α) =HW(λ, β, t, α)∩HW.

When we take β = 0, the class of HW(λ, β, t, α) reduces to the class of harmonic functions HG(λ, t, α)≡HW(λ,0, t, α) as follows

<

(1−t)D λ_f(z)

z +t

(Dλf(z))0 z0

≥α. (12)

(4)

<

(1 +t)(D λ_f_(z))0

z0 +t

(Dλ_f_(z))00 z00 −2t

≥α. (13)

2. Coefficient Bounds

Now, firstly we presented sufficient coefficient conditions for functions to be in the class of HW(λ, β, t, α). Furthermore, it is shown that these conditions are also necessary for the functions ofHW(λ, β, t, α).

Theorem 2.1. Let f =h+g, h and g be given by (1) and

∞

X

n=2

(β+t)(n−1) +βt(1 +n2) + 1

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) |an|

+ ∞

X

n=1

(β+t)(n+ 1)−βt(1 +n2)−1

Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |bn| ≤1−α.

(14)

Then f(z)∈HW(λ, β, t, α).

Proof. Using the fact that Rew≥α if and only if |w+ 1−α| ≥ |w−1−α|

w= (1−β)(1−t)D λ_f_(z)

z + (β+t)

(Dλf(z))0 z0 +βt

(Dλf(z))00 z00 −2βt

It is enough to show that |w+ 1−α| − |w−1−α| ≥0.

Now, we have

|w+1−α|=

(1−β)(1−t) 1 + ∞

X

n=2

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) anz

n−1₊

∞

X

n=1

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) bn(z)

n−1

!

+(β+t) 1 + ∞

X

n=2

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) nanz

n−1₋

∞

X

n=1

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) nbn(z)

n−1

!

+βt 1 + ∞

X

n=2

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) nanz

n−1₊

∞

X

n=2

Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) n(n−1)anz n−1

+ ∞

X

n=1

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) nbn(z)

n−1₊

∞

X

n=1

Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) n(n−1)bn(z) n−1

!

−2βt+ 1−α

≥2−α−

∞

X

n=2

|1 + (β+t)(n−1) +βt(1 +n2)|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |an||z n−1_|

−

∞

X

n=1

|1−(β+t)(n+ 1) +βt(1 +n2)|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |bn||z n−1_|

and

|w−1−α|=

(1−β)(1−t) 1 + ∞

X

n=2

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) anz

n−1₊

∞

X

n=1

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) bn(z)

n−1

(5)

+(β+t) 1 + ∞

X

n=2

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) nanz

n−1₋

∞

X

n=1

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) nbn(z)

n−1

!

+βt 1 + ∞

X

n=2

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) nanz

n−1₊

∞

X

n=2

Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) n(n−1)anz n−1

+ ∞

X

n=1

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) nbn(z)

n−1₊

∞

X

n=1

Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) n(n−1)bn(z) n−1

!

−2βt−1−α

≤α+ ∞

X

n=2

|1 + (β+t)(n−1) +βt(1 +n2)|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |an||z n−1_|

+ ∞

X

n=1

|1−(β+t)(n+ 1) +βt(1 +n2)|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |bn||z n−1_|_.

So by using (14) we have

|w+1−α|−|w−1−α| ≥2

"

1−α−

∞

X

n=2

(β+t)(n−1) +βt(1 +n2) + 1

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) |an|

−

∞

X

n=1

(β+t)(n+ 1)−βt(1 +n2)−1

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) |bn|

#

≥0

and so the proof is completed.

Remark 1 The coefficient bound (14) in previous theorem is sharp for the function

f(z) =z+ ∞

X

n=2

un

|(β+t)(n−1) +βt(1 +n2_{) + 1}_|

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) z

n

∞

X

n=1

wn

|(β+t)(n+ 1)−βt(1 +n2₎₋₁_|

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) (z)

n_, ₍₁₅₎

where

1 1−α

∞

X

n=2 |un|+

∞

X

n=1 |wn|

!

= 1.

Theorem 2.2. Let f =h+g∈HW . f(z)∈HW(λ, β, t, α) if and only if

∞

X

n=2

(β+t)(n−1) +βt(1 +n2) + 1

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) |an|

+ ∞

X

n=1

(β+t)(n+ 1)−βt(1 +n2)−1

Γ(2−λ)Γ(n+ 1)

(6)

Proof. From Theorem 2.1 and since HW(λ, β, t, α) ⊂ HW(λ, β, t, α) we conclude the ”if” part. For the ”only if” part, assume that f(z) ∈ HW(λ, β, t, α). Therefore, for z=reiθ ∈U, we have

<

(1−β)(1−t)D λ_f_(z)

z + (β+t)

(Dλf(z))0 z0 +βt

(Dλf(z))00 z00 −2βt

=<

(

(1−β)(1−t) 1 + ∞

X

n=2

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) anz

n−1₊

∞

X

n=1

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) bn(z)

n−1

!

+(β+t) 1−

∞

X

n=2

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) nanz

n−1₋

∞

X

n=1

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) nbn(z)

n−1

!

+βt 1−

∞

X

n=2

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) nanz

n−1₋

∞

X

n=2

Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) n(n−1)anz n−1

+ ∞

X

n=1

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) nbn(z)

n−1₋

∞

X

n=1

Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) n(n−1)bn(z) n−1

!

−2βt

)

=<

(

1−

∞

X

n=2

[(1−β)(1−t) + (β+t)n+nβt+n(n−1)βt]Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) anz

n−1

+ ∞

X

n=1

[(1−β)(1−t)−(β+t)n+nβt+n(n−1)βt]Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) bn(z)

n−1

)

≥1−

∞

X

n=2

|(β+t)(n−1) +βt(1 +n2) + 1|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |an|r n−1

−

∞

X

n=1

|(β+t)(n+ 1)−βt(1 +n2)−1|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |bn|r

n−1 _≥_α.

The last inequality holds for all z∈ U. So if z =r → 1 we obtain the required result given by (16). So the proof of the Theorem 2.2 is completed.

As special cases of Theorem 2.2, we can obtain the following two corollaries:

Corollary 2.1. f =h+g∈HG(λ, t, α)≡HG(λ, t, α)∩HW if and only if

∞

X

n=2

|t(n−1) + 1|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |an|+ ∞

X

n=1

|t(n+ 1)−1|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |bn| ≤1−α.

Especially for λ= 0 ∞

X

n=2

|t(n−1) + 1||an|+ ∞

X

n=1

(7)

Corollary 2.2. f =h+g∈HA(λ, t, α)≡HA(λ, t, α)∩HW if and only if

∞

X

n=2

|n(1 +t+nt)|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |an|+ ∞

X

n=1

|n(1 +t−nt)|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |bn| ≤1−α.

Especially for λ= 0 ∞

X

n=2

|n(1 +t+nt)||an|+ ∞

X

n=1

|n(1 +t−nt)||bn| ≤1−α.

3. Extreme Points

In the following theorem, we represent extreme points of HW(λ, β, t, α).

Theorem 3.1. f =h+g∈HW(λ, β, t, α) if and only if it can be expressed as

f(z) =X1z+

∞

X

n=2

Xnhn(z) + ∞

X

n=1

Yngn(z), z∈U (17)

where

hn(z) =z−

1−α

|(β+t)(n−1) +βt(1 +n2_{) + 1}_|

Γ(n+ 1−λ) Γ(2−λ)Γ(n+ 1)z

n _(n_{= 2,}_{3, ...) (18)}

gn(z) =z+

1−α

|(β+t)(n+ 1)−βt(1 +n2₎₋₁_|

Γ(n+ 1−λ) Γ(2−λ)Γ(n+ 1)(z)

n _(n_{= 1,}_{2, ...) (19)}

X1 ≥0, Y1≥0, X1z+

∞

X

n=2

Xn+ ∞

X

n=1

Yn= 1 Xn≥0, Yn≥0 f or n= 2,3, ... .

Proof. Iff(z) be given by (17), then

f(z) =z−

∞

X

n=2

1−α

|(β+t)(n−1) +βt(1 +n2_{) + 1}_|

Γ(n+ 1−λ) Γ(2−λ)Γ(n+ 1)Xnz

n

+ ∞

X

n=1

1−α

|(β+t)(n+ 1)−βt(1 +n2₎₋₁_|

Γ(n+ 1−λ)

Γ(2−λ)Γ(n+ 1)Yn(z) n_.

Since by (16), we have

∞

X

n=2

|(β+t)(n−1) +βt(1 +n2) + 1| 1−α

|(β+t)(n−1) +βt(1 +n2_{) + 1}_|

Γ(n+ 1−λ) Γ(2−λ)Γ(n+ 1)|Xn|

+ ∞

X

n=1

|(β+t)(n+ 1)−βt(1 +n2)−1| 1−α

|(β+t)(n+ 1)−βt(1 +n2₎₋₁_|

Γ(n+ 1−λ) Γ(2−λ)Γ(n+ 1)|Yn|

= (1−α) ∞

X

n=2 |Xn|+

∞

X

n=1 |Yn|

!

(8)

So f(z)∈HW(λ, β, t, α). Conversely, assume f(z)∈HW(λ, β, t, α) by setting

X1 = 1−

∞

X

n=2 |Xn|+

∞

X

n=1 |Yn|

!

, where X1 ≥0. T hen

Xn=

|(β+t)(n−1) +βt(1 +n2) + 1|

1−α

Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |an|, (n≥2)

Yn=

|(β+t)(n+ 1)−βt(1 +n2)−1|

1−α

Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |bn| (n≥1),

we obtain

f(z) =z−

∞

X

n=2

|an|zn+ ∞

X

n=1

|bn|(z)n

= ∞

X

n=2

(1−α)Xn

|(β+t)(n−1) +βt(1 +n2_{) + 1}_|

Γ(n+ 1−λ) Γ(2−λ)Γ(n+ 1)z

n

+ ∞

X

n=1

(1−α)Yn

|(β+t)(n+ 1)−βt(1 +n2₎₋₁_|

Γ(n+ 1−λ) Γ(2−λ)Γ(n+ 1)(z)

n

=

"

1−

∞

X

n=2

Xn+ ∞

X

n=1

Yn

!#

z+ ∞

X

n=2

hn(z)Xn+ ∞

X

n=1

gn(z)Yn

=X1z+

∞

X

n=2

Xnhn(z)+ ∞

X

n=1

Yngn(z),

that is the required representation.

4. Convex Combination

Now we introduce HW(λ, β, t, α) is closed under convex combination.

Theorem 4.1. If fni(i= 1,2, ...) belongs to HW(λ, β, t, α), then the function

φ(z) = ∞

X

i=1

σifni(z) is also in HW(λ, β, t, α), where fni(z) is def ined by

fni(z) =z−

∞

X

n=2

aniz

n₊ ∞

X

n=1

bni(z)

n _(i_{= 1,}_{2, ...,} ₀_≤_σ i≤1,

∞

X

i=1

σi= 1). (20)

Proof. Since fni(z)∈HW(λ, β, t, α), by (16) we have

∞

X

n=2

|(β+t)(n−1)+βt(1+n2)+1|Γ(2−λ)Γ(n+ 1)

(9)

+ ∞

X

n=1

|(β+t)(n+1)−βt(1+n2)−1|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |bni| ≤1−α, (i= 1,2, ...).

Also

φ(z) = ∞

X

i=1

σifi(z) =z− ∞

X

n=2

∞

X

i=1

σiani !

zn+ ∞

X

n=1

∞

X

i=1

σibni !

(z)n.

Now according to Theorem 2.2 we have

∞

X

n=2

|(β+t)(n−1) +βt(1 +n2) + 1|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ)

∞

X

i=1

σiani

+ ∞

X

n=1

|(β+t)(n+1)−βt(1+n2)−1|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ)

∞

X

i=1

σibni

= ∞

X

i=1

σi

( _∞

X

n=2

|(β+t)(n−1) +βt(1 +n2) + 1|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |ani|

+ ∞

X

n=1

|(β+t)(n+ 1)−βt(1 +n2)−1|Γ(2−λ)Γ(n+ 1)

Γ(n+ 1−λ) |bn,i|

)

≤(1−α) ∞

X

i=1

σi = 1−α.

Thus φ(z)∈HW(λ, β, t, α).

So, we note that HW(λ, β, t, α) is a convex set.

5. Distortion Bounds

In the following theorem we obtain distortion bounds for f ∈HW(λ, β, t, α).

Theorem 5.1. Let f =h+g∈HW(λ, β, t, α), β ≥1, |z|=r <1. Then, we have

|f(z)| ≥(1− |b1|)r−

1−α

1 +β+t(1 + 5β) −

−1 + 2(β+t−βt) 1 +β+t(1 + 5β) |b1|

2−λ 2 r

2_, ₍₂₁₎

and

|f(z)| ≤(1 +|b1|)r+

1−α

1 +β+t(1 + 5β) −

−1 + 2(β+t−βt) 1 +β+t(1 + 5β) |b1|

2−λ 2 r

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Proof. Assume f(z)∈HW(λ, β, t, α), then by (16), we have

f(z) =

z−

∞

X

n=2

|an|zn+ ∞

X

n=1

|bn|(z)n

=

z+|b1|(z)−

∞

X

n=2

(|an|zn− |bn|(z)n)

≥(1− |b1|)r−

∞

X

n=2

(|an|+|bn|)r2

≥(1− |b1|)r−

1−α 1 +β+t(1 + 5β).

2−λ 2 ×

∞

X

n=2

1 +β+t(1 + 5β) 1−α

2

2−λ(|an|+|bn|)r

2

≥(1− |b1|)r−

1−α 1 +β+t(1 + 5β).

2−λ 2 ×

∞

X

n=2

(β+t)(n−1) +βt(1 +n2_{) + 1}

1−α

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) |an|

+(β+t)(n+ 1)−βt(1 +n

2₎₋₁

1−α

Γ(2−λ)Γ(n+ 1) Γ(n+ 1−λ) |bn|

r2

≥(1− |b1|)r−

1−α 1 +β+t(1 + 5β).

2−λ

2 ×

1−−1 + 2(β+t−βt)

1−α |b1|

r2

=r− |b1|r−

1−α

1 +β+t+ 5βt−

−1 + 2(β+t−βt) 1 +β+t+ 5βt |b1|

2−λ 2 r

2_.

Relation (22) can be proved by using the similar statements. Therefore, it is omitted. So the proof is completed.

Corollary 5.1. If f ∈HG(λ, t, α) then

|f(z)| ≥(1− |b1|)r−

1−α t+ 1 −

2t−1 t+ 1|b1|

2−λ 2 r

2

and

|f(z)| ≤(1 +|b1|)r+

1−α t+ 1 −

2t−1 t+ 1|b1|

2−λ 2 r

2_.

Especially for λ= 0, we obtain inequalities as follows

|f(z)| ≥(1− |b1|)r−

1−α t+ 1 −

2t−1 t+ 1|b1|

r2 and

|f(z)| ≤(1 +|b1|)r+

1−α t+ 1 −

2t−1 t+ 1 |b1|

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Corollary 5.2. If f ∈HA(λ, t, α) then

|f(z)| ≥(1− |b1|)r−

1−α 2(1 + 3t) −

1

2(1 + 3t)|b1|

2−λ 2 r

2

and

|f(z)| ≤(1 +|b1|)r+

1−α 2(1 + 3t) −

1

2(1 + 3t)|b1|

2−λ 2 r

2_.

Especially for λ= 0 , we obtain inequalities as follows

|f(z)| ≥(1− |b1|)r−

1−α 2(1 + 3t) −

1

2(1 + 3t)|b1|

r2 and

|f(z)| ≤(1 +|b1|)r+

1−α 2(1 + 3t) −

1

2(1 + 3t)|b1|

r2.

AcknowledgementThe work presented here was partially supported by Batman Uni-versity Scientific Research Coordination Unit. Project Number: BTBAP-2016-14B102.

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F. M¨uge Sakarwas born in Diyarbakır, Turkey. She graduated from Di-cle University in 2001. She also received her M.Sc.(2008) and Ph.D.(2012) degrees from Dicle University (Diyarbakır). She has been working as an assistant professor in Faculty of Economics and Administrative Sciences at Batman University, Batman, Turkey since 2013. Her research area includes Geometric Function Theory, Analytic Functions and Bi-univalent Functions, especially Coefficient Bounds.

Yasemin Ba˘gcıwas born in 1989. She has still been working as a Mathe-matics teacher in Batman Z¨ubeyde Hanım Vocational and Technical Anato-lian High School, Turkey since 2012. She is still an M.Sc student at Batman University, Institude of Science, Master’s Degree in Mathematics, Batman, Turkey. Her research interest includes Univalent Functions, Harmonic Func-tions.