Argument estimates of certain multivalent functions involving a linear operator

(1)

http://ijmms.hindawi.com © Hindawi Publishing Corp.

ARGUMENT ESTIMATES OF CERTAIN MULTIVALENT

FUNCTIONS INVOLVING A LINEAR OPERATOR

NAK EUN CHO, J. PATEL, and G. P. MOHAPATRA

Received 28 December 2001

The purpose of this paper is to derive some argument properties of certain multivalent functions in the open unit disk involving a linear operator. We also investigate their integral preserving property in a sector.

2OOO Mathematics Subject Classiﬁcation: 30C45.

1. Introduction. LetᏭpdenote the class of functions of the form

f (z)=zp+

∞

n=1

an+pzn+p p∈N= {1,2, . . .} (1.1)

which are analytic in the open unit diskᐁ= {z:|z|<1}. A functionf∈Ꮽpis said to bep-valently starlike of orderαinᐁ, if it satisﬁes

Re

zf(z) f (z)

> α (0≤α < p; z∈ᐁ). (1.2)

We denote this class by᏿∗p(α). A functionf∈Ꮽpis said to bep-valently convex of orderαinᐁ, if it satisﬁes

Re

1+zf(z) f(z)

> α (0≤α < p;z∈ᐁ). (1.3)

The class ofp-valently convex functions of orderαis denoted by᏷p(α). It follows from (1.2) and (1.3) that

f∈᏷p(α)⇐⇒ zf

p ∈᏿p(α). (1.4)

Further, a functionf∈Ꮽpis said to bep-valently close-to-convex of orderβand type α, if there exists a functiong∈᏿∗

p(α)such that

Re

zf(z) g(z)

> β (0≤α, β < p;z∈ᐁ). (1.5)

It is well known (see [10]) that everyp-valently close-to-convex function isp-valent inᐁ.

For arbitrary ﬁxed real numbersAandB (−1≤B < A≤1), letᏼ(A, B)denote the class of functions of the form

(2)

which are analytic inᐁand satisﬁes the condition

φ(z)≺1₁+₊Az_Bz (z∈ᐁ), (1.7)

where the symbol≺stands for subordination. The classᏼ(A, B)was introduced and studied by Janowski [8].

We note that a functionφ∈ᏼ(A, B), if and only if

φ(z)−1₁−₋AB_B2

< A−B

1−B2 (B≠−1, z∈ᐁ), Reφ(z)>1−A

2 (B= −1, z∈ᐁ).

(1.8)

For a function f ∈Ꮽ, given by (1.1), the generalized Bernardi-Libera-Livingston integral operatorF [1] is deﬁned by

F (z)=γ+p zγ

z

0t

γ−1_{f (t) dt}

=zp₊ ∞

n=1 γ+p

γ+p+nan+pz

n+p _{(γ >}₋_p_; _z_∈_ᐁ_). (1.9)

It readily follows from (1.9) that

f∈Ꮽp ⇒F∈Ꮽp. (1.10)

Let

φp(a, c;z)= ∞

n=0 (a)n (c)n

zn+p (c≠0,−1,−2, . . .; z∈ᐁ), (1.11)

and we deﬁne a linear operatorLp(a, c)onᏭpby

Lp(a, c)f (z)=φp(a, c;z)∗f (z) (z∈ᐁ), (1.12)

where(x)n=Γ(n+x)/Γ(x)and the symbol∗is the Hadamard product or convolu-tion. Clearly,Lp(a, c)mapsᏭp into itself. Further,Lp(a, a) is the identity operator and

Lp(a, c)=Lp(a, b)Lp(b, c)=Lp(b, c)Lp(a, b) (b, c≠0,−1,−2, . . .). (1.13)

Thus, ifa≠0,−1,−2, . . . ,thenLp(a, c)has an inverseLp(c, a). We also observe that forf∈Ꮽp,

Lp(p+1, p)f (z)= zf(z)

p , Lp(µ+p,1)f (z)=D

µ+p−1_{f (z),} _(1.14)

(3)

L(a, c)introduced by Carlson and Shaﬀer [3] in their systemic investigation of certain classes of starlike, convex, and prestarlike hypergeometric functions.

In the present paper, we give some argument properties of certain class of analytic functions inᏭp involving the linear operatorLp(a, c). An application of a certain integral operator is also considered. The results obtained here, besides extending the works of Bulboac˘a [2], Chichra [4], Cho et al. [5], Fukui et al. [6], Libera [9], Nunokawa [13], and Sakaguchi [16], it yields a number of new results.

2. Main results. To establish our main results, we need the following lemmas.

Lemma2.1[11]. Leth(z)be convex (univalent) inᐁand letψ(z)be analytic inᐁ withRe{ψ(z)} ≥0. Ifφ(z)is analytic inᐁandφ(0)=ψ(0), then

φ(z)+ψ(z)zφ(z)≺h(z) (z∈ᐁ) (2.1)

implies

φ(z)≺h(z) (z∈ᐁ). (2.2)

Lemma2.2[12]. Letφ(z)be analytic inᐁ,φ(0)=1, φ(z)≠0inᐁand suppose that there exists a pointz0∈ᐁsuch that

_arg_φ(z)_<π 2η

|z|<z0, _arg_φ

z0= π 2η,

(2.3)

whereη >0. Then

z0φ

z0

φz0

=ikη, (2.4)

where

k≥1₂

d+1_d

whenargφz0

=π₂η,

k≤ −1₂

d+1_d

when argφz0

= −π₂η,

(2.5)

where

φz01/η= ±id (d >0). (2.6) We now derive the following theorem.

Theorem_2.3. _Let_{a >}_0,−₁≤_{B < A}≤_1,_f∈Ꮽ_{p, and suppose that}_g∈Ꮽ_p_satisﬁes

Lp(a+1, c)g(z) Lp(a, c)g(z) ≺

1+Az

1+Bz (z∈ᐁ). (2.7)

If

arg

(1−λ)Lp(a, c)f (z) Lp(a, c)g(z)+

λLp(a+1, c)f (z) Lp(a+1, c)g(z)−

β

<π

2δ (λ≥0; 0≤β <1; 0< δ≤1; z∈ᐁ),

(4)

then

arg

Lp(a, c)f (z) Lp(a, c)g(z)−

β<π

2η (z∈ᐁ), (2.9)

whereη (0< η≤1)is the solution of the equation

δ=

      

η+_π2tan−1

ληsin(π /2)1−t(A, B)

a(1+A)/(1+B)+ληcos(π /2)1−t(A, B)

, forB≠−1,

η, forB= −1,

(2.10)

when

t(A, B)=_π2sin−1 _A₋_B

1−AB

. (2.11)

Proof. _Let

Lp(a, c)f (z) Lp(a, c)g(z)=

β+(1−β)φ(z). (2.12)

Thenφ(z)is analytic inᐁwithφ(0)=1. On diﬀerentiating both sides of (2.12) and using the identity

zLp(a, c)f (z)=aLp(a+1, c)f (z)−(a−p)Lp(a, c)f (z) (2.13)

in the resulting equation, we deduce that

(1−λ)Lp(a, c)f (z) Lp(a, c)g(z)+

λLp(a+1, c)f (z) Lp(a+1, c)g(z)−

β=(1−β)

φ(z)+λzφ(z) ar (z)

, (2.14)

where

r (z)=Lp(a+1, c)g(z) Lp(a, c)g(z)

. (2.15)

If we let

r (z)=ρe(π θ/2)i, (2.16)

then from (2.7) followed by (1.8), it follows that

1−A 1−B < ρ <

1+A 1+B,

−t(A, B) < θ < t(A, B) forB≠−1,

(2.17)

whent(A, B)is given by (2.11), and

1−A

2 < ρ <∞,

−1< θ <1 forB= −1.

(5)

Leth(z)be the function which maps onto the angular domain{w:|arg{w}|< (π /2)δ}

withh(0)=1. ApplyingLemma 2.1for thish(z)withψ(z)=λ/(ar (z)), we see that Reφ(z) >0 inᐁand henceφ(z)≠0 inᐁ.

If there exists a point z0∈ ᐁ such that conditions (2.3) are satisﬁed, then by

Lemma 2.2we obtain (2.4) under restrictions (2.5) and (2.6).

At ﬁrst, suppose thatp(z0)1/η=id (d >0). For the caseB≠−1, we obtain

arg

(1−λ)Lp(a, c)f

z0

Lp(a, c)g

z0

+λLp(a+1, c)f

z0

Lp(a+1, c)g

z0 −β

=argφz0+arg

1+ λ arz0

z0φ

z0

φz0

=π₂η+arg

1+iηkλe− (π θ/2)i

ρa

=π₂η+tan−1

ληksin(π /2)(1−θ) ρa+ληkcos(π /2)(1−θ)

≥π₂η+tan−1

a(1+A)/(1+B)+ληcos(π /2)1−t(A, B)

≥π

2δ,

(2.19)

whereδandt(A, B)are given by (2.10) and (2.11), respectively. Similarly, for the case B= −1, we have

arg

(1−λ)Lp(a, c)f

z0 Lp(a, c)gz0+

λLp(a+1, c)f

z0 Lp(a+1, c)gz0−

β

≥π₂η. (2.20)

This is a contradiction to the assumption of our theorem.

Next, suppose thatφ(z0)1/η= −id (d >0). For the caseB≠−1, applying the same method as above, we have

arg

(1−λ)Lp(a, c)f

z0 Lp(a, c)gz0+

λLp(a+1, c)f

z0 Lp(a+1, c)gz0−

β

≤ −π₂η−tan−1

a(1+A)/(1+B)+ληcos(π /2)1−t(A, B)

≤ −π

2δ,

(2.21)

whereδandt(A, B)are given by (2.10) and (2.11), respectively and for the caseB= −1, we have

arg

(1−λ)Lp(a, c)f

z0 Lp(a, c)gz0+

λLp(a+1, c)f

z0 Lp(a+1, c)gz0−

β

≤ −π₂η (2.22)

(6)

Remark2.4. Fora=c=p,A=1,B= −1, andλ=1,Theorem 2.3is the recent result obtained by Nunokawa [13].

Takinga=µ+p (µ >−p), c=1, A=1, andB=0 inTheorem 2.3, we have the following corollary.

Corollary2.5. Iff∈Ꮽ_psatisﬁes

arg

(1−λ)D

µ+p−1_{f (z)} Dµ+p−1_g(z)+λ

Dµ+p_{f (z)} Dµ+p_g(z)−β

<π

2δ (λ≥0; 0< δ≤1; 0≤β <1;z∈ᐁ)

(2.23)

for someg∈Ꮽpsatisfying the condition

_DDµµ++pp−g(z)1_g(z)−1

< α (0< α≤1; z∈ᐁ), (2.24)

then

arg

Dµ+p−1f (z) Dµ+p−1_g(z)

<π2η (z∈ᐁ), (2.25)

δ=η+_π2tan−1

ληsinπ /2−sin−1α (µ+p)(1+α)+ληcosπ /2−sin−1α

. (2.26)

LettingB→A (A <1)andg(z)=zp_in_{Theorem 2.3}_{, we get the following corollary.}

Corollary_2.6. _If_f∈Ꮽ_p_satisﬁes

arg

(1−λ)Lp(a, c)f (z) zp +λ

Lp(a+1, c)f (z)

zp −β

<π

2δ (a >0; λ≥0; 0≤β <1; 0< δ≤1; z∈ᐁ),

(2.27)

then

arg

Lp(a, c)f (z) zp −β

<

π

2η (z∈ᐁ), (2.28)

δ=η+_π2tan−1 _λη

a

. (2.29)

Corollary2.7. Under the hypothesis ofCorollary 2.6, we have argH(z)−β<π

(7)

where the functionH(z)is deﬁned inᐁby

H(z)= z

0

Lp(a, c)f (t)

tp dt (2.31)

andη (0< η≤1)is the solution of (2.29).

Remark2.8. Takinga=c=p,λ=1, andβ=0 inCorollary 2.6, a=c=p and β=0 inCorollary 2.7, we get the corresponding results obtained by Cho et al. [5].

SettingA=1−(2α/p) (0≤α < p),B= −1, andδ=1 inTheorem 2.3, we have the following corollary.

Corollary_2.9. _Let_{a >}_0,_f∈Ꮽ_{p, and}_g∈᏿∗_p_(α)_{. If}

Re

(1−λ)Lp(a, c)f (z) Lp(a, c)g(z)+

λLp(a+1, c)f (z) Lp(a+1, c)g(z)

> β (λ≥0; 0≤β <1;z∈ᐁ), (2.32)

then

Re

Lp(a, c)f (z) Lp(a, c)g(z)

> β (z∈ᐁ). (2.33)

Remark2.10. Fora=c=p=1 andα=0,Corollary 2.9is the result by Bulboac˘a [2]. If we puta=c=p=1,β=0, andg(z)=zinCorollary 2.9, then we have the result due to Chichra [4]. Further, takinga=c=p,λ=1, andα=β=0 inCorollary 2.9, we get the corresponding results of Libera [9] and Sakaguchi [16].

Theorem2.11. Iff∈Ꮽ_psatisﬁes

arg

Lp(a, c)f (z) zp −β

<π2δ (0≤β <1; 0< δ≤1; z∈ᐁ), (2.34)

then arg

(γ+p)0ztγ−1Lp(a, c)f (t) dt

zγ+p −β

<

π

2η (0< γ+p; z∈ᐁ), (2.35)

δ=η+_π2tan−1

η γ+p

. (2.36)

Proof. _{Consider the function}_φ(z)_{deﬁned in}ᐁ_by

(γ+p)₀ztγ−1Lp(a, c)f (t) dt

zγ+p =β+(1−β)φ(z). (2.37)

Thenφ(z) is analytic inᐁ with φ(0)=1. Diﬀerentiating both sides of (2.37) and simplifying, we get

Lp(a, c)f (z)

zp −β=(1−β)

φ(z)+zφ _(z)

γ+p

. (2.38)

(8)

Takinga=p+1,c=p,β=ρ/p, andδ=1 inTheorem 2.11, we have the following corollary.

Re _f_(z)

zp−1

> ρ (0≤ρ < p; z∈ᐁ), (2.39)

then

arg

(γ+p)0ztγ−1f(t) dt

zγ+p −ρ

<

π

2η (z∈ᐁ), (2.40)

η+2 πtan

−1 η γ+p

=1. (2.41)

Theorem_2.13. _If_f∈Ꮽ_p_satisﬁes

arg

Lp(a+1, c)f (z) Lp(a, c)f (z) −

a−p−γ a

<π2δ (a >0; p+γ >0; 0< δ≤1;z∈ᐁ), (2.42)

then

arg

zγ_L

p(a, c)f (z) z

0tγ−1Lp(a, c)f (t) dt

<

π

2η (z∈ᐁ), (2.43)

whereη (0< η≤1)is the solution of (2.36).

Proof. _{Our proof of}_{Theorem 2.13}_{is much akin to that of}_{Theorem 2.3}_{. Indeed,} in place of (2.37), we deﬁne the functionφ(z)by

φ(z)= zγLp(a, c)f (z) (γ+p)0ztγ−1Lp(a, c)f (t) dt

(z∈ᐁ), (2.44)

and applyLemma 2.1(withψ(z)=1/(γ+p)) as before. We choose to skip the details involved.

Settinga=c=pandδ=1 inTheorem 2.13, we obtain the following corollary.

Re

zf(z) f (z)

>−γ (γ+p >0; z∈ᐁ), (2.45)

then

arg

zγ_{f (z)} z

0tγ−1f (t) dt

<π2η (z∈ᐁ), (2.46)

(9)

Re

1+zf_f_(z)(z)

>−γ (γ+p >0;z∈ᐁ), (2.47)

then

arg

zf(z) f (z)−(γ/zγ₎z

0tγ−1f (t) dt

<

π

2η (z∈ᐁ), (2.48)

By settingγ=0 inCorollary 2.15, we have the following corollary.

Corollary2.16. Iff∈᏷_p(0), then

arg

zf(z)

f (z)

<π2η (z∈ᐁ), (2.49)

whereη (0< η≤1)is the solution of the equation:

η+_π2tan−1 _η

p

=1. (2.50)

Similarly, we have the following theorem.

Theorem_2.17. _If_f∈Ꮽ_p_satisﬁes

arg

Lp(a+1, c)f (z) Lp(a, c)f (z) −

β

<

π

2δ (a >0; 0≤β <1; 0< δ≤1;z∈ᐁ), (2.51)

then

arg

Lp(a, c)f (z) zp

<

π

2η (z∈ᐁ), (2.52)

δ= 2 πtan

−1 η (1−β)a

. (2.53)

Theorem_2.18. _Let_f∈Ꮽ_p_{and suppose that}

B < A≤B+p(1−B)

a (a >0; −1≤B < A≤1). (2.54)

If arg

(1−λ)Lp(a+1, c)f (z) Lp(a, c)g(z) +

λ

Lp(a+1, c)f (z)

Lp(a, c)g(z) −β

<π

2δ (λ≥0; 0≤β <1; 0< δ≤1; z∈ᐁ),

(10)

for someg∈Ꮽpsatisfying

Lp(a+1, c)g(z) Lp(a, c)g(z) ≺

1+Az

1+Bz (z∈ᐁ), (2.56)

then

arg

Lp(a+1, c)f (z) Lp(a, c)g(z) −

β

<π2η (z∈ᐁ), (2.57)

δ=

      

η+2 πtan

−1 ληsin(π /2)

1−t(A, B)

p(1+B)+a(A−B)/(1+B)+ληcos(π /2)1−t(A, B)

, forB≠−1,

η, forB=−1,

(2.58)

when

t(A, B)=_π2sin−1

a(A−B) p1−B2₋_aB(A₋_B)

. (2.59)

Proof. _Let

Lp(a+1, c)f (z) Lp(a, c)g(z) =

β+(1−β)φ(z), r (z)=Lp(a+1, c)g(z) Lp(a, c)g(z)

, (2.60)

we have

(1−λ)Lp(a+1, c)f (z) Lp(a, c)g(z) +

λ

Lp(a+1, c)f (z)

Lp(a+1, c)g(z)

−β=(1−β)

φ(z)+ λzφ _(z)

ar (z)+p−a

.

(2.61)

The remaining part of the proof ofTheorem 2.18is similar to that ofTheorem 2.3. So we omit the details.

Puta=c=p,λ=1,A=α/p, andB=0 inTheorem 2.18, we have the following corollary.

arg

zf(z) g_(z) −β

<

π

2δ (0≤β < p; 0< δ≤1; z∈ᐁ), (2.62)

zg(z)

g(z) −p

(11)

then

arg

zf(z)

g(z) −β

<π2η (z∈ᐁ), (2.64)

δ=η+_π2tan−1

ηsinπ /2−sin−1(α/p) p+α+ηcosπ /2−sin−1(α/p)

. (2.65)

Lemma2.20. Let

α=ξ+ ξ γ+p+aξ

0≤(a−1)/a < ξ < α <1 (2.66)

and the functionG(z)be deﬁned by

G(z)=γ_z+_γp z

0

tγ−1g(t) dt g∈Ꮽp

(2.67)

forγ > (aξ2₊_(p₊₁₋_a)ξ₋_p)/(₁₋_ξ)_{. If}_g_∈_Ꮽ

psatisﬁes

Lp(a+1, c)g(z) Lp(a, c)g(z) −

1< α (z∈ᐁ), (2.68)

then

Lp(a+1, c)G(z) Lp(a, c)G(z) −

1< ξ (z∈ᐁ). (2.69)

Proof. Deﬁning the functionw(z)by

Lp(a+1, c)G(z) Lp(a, c)G(z) =

1+ξw(z), (2.70)

we see thatw(z)is analytic inᐁwithw(0)=0. Now, using the identities

zLp(a, c)G(z)=aLp(a+1, c)G(z)−(a−p)Lp(a, c)G(z), (2.71)

zLp(a, c)G(z)

₌_(γ₊_p)L

p(a, c)g(z)−γLp(a, c)G(z) (2.72)

in (2.70), we get

Lp(a, c)G(z) Lp(a, c)g(z)=

γ+p

γ+p+aξw(z). (2.73)

Making use of the logarithmic diﬀerentiation of both sides of (2.73) and using identity (2.71) for bothg(z)andf (z)in the resulting equation, we deduce that

Lp(a+1, c)g(z) Lp(a, c)g(z) −

1=ξw(z)+ zw _(z)

γ+p+aξw(z)

(12)

We assume that there exists a pointz0∈ᐁsuch that max|z|<|z0||w(z)| = |w(z0)| =1. Then by Jack’s lemma [7], we havez0w(z0)=kw(z0) (k≥1). Letw(z0)=eiθ, and apply this result tow(z)atz0∈ᐁ, we get

Lp(a+1, c)gz0 Lp(a, c)g

z0

−1=ξ1+ k γ+p+aξeiθ

=ξ

(γ+p+k)2₊₂_aξ(γ₊_p₊_k)_cos_θ₊_(aξ)2 (γ+p)2₊₂_aξ(γ₊_p)_cos_θ₊_(aξ)2

1/2 .

(2.75)

Since the right side of (2.75) is decreasing for 0≤θ <2πandγ >{aξ2₊_(p₊₁₋_a)ξ₋ p}/(1−ξ), we obtain

Lp(a+1, c)gz0 Lp(a, c)gz0 −

1≤ξ(γ+p+1+aξ)

γ+p+aξ , (2.76)

which contradicts our hypothesis and hence we get

w(z)=_ξ1Lp_L(a+1, c)G(z)

p(a, c)G(z) −

1<1 (z∈ᐁ). (2.77)

This completes the proof ofLemma 2.20.

Remark2.21. We note that fora=c=p=1,Lemma 2.20yields the correspond-ing result obtained by Fukui et al. [6].

Theorem _2.22. _Let_α _{be as given in (2.66) and}_γ∗ _>_max{_(aξ2+_(p+₁−_a)ξ− p)/(1−ξ), aξ−p}. Iff∈Ꮽpsatisﬁes

arg

Lp(a+1, c)f (z) Lp(a, c)g(z) −

β

<π2δ (0≤β <1; 0< δ≤1; z∈ᐁ), (2.78)

for somef∈Ꮽpsatisfying condition (2.68), then

arg

Lp(a+1, c)F (z) Lp(a, c)G(z) −

β

<π2η (z∈ᐁ), (2.79)

where the functionF (z)andG(z)are deﬁned forγ∗by (1.9) and (2.67), respectively andη (0< η≤1)is the solution of the equation

δ=η+2 πtan

−1

ηsinπ /2−sin−1aξ/γ∗+p γ∗+p+aξ+ηcosπ /2−sin−1aξ/γ∗+p

. (2.80)

Proof. _{Consider the function}_φ(z)_{deﬁned in}ᐁ_by

Lp(a+1, c)F (z) Lp(a, c)G(z) =

(13)

Thenφ(z)is analytic inᐁwithφ(0)=1. Taking logarithmic diﬀerentiation on both sides of (2.81) and using identity (2.71) in the resulting equation, we get

zLp(a+1, c)F (z) Lp(a+1, c)F (z) =

p−a+aLp(a+1, c)G(z) Lp(a, c)G(z) +

(1−β) zφ(z)

β+(1−β)φ(z). (2.82)

From the deﬁnition ofF (z), we have

γ∗+pLp(a, c)f (z)=a

Lp(a+1, c)F (z) ₊_γ∗_L

p(a+1, c)F (z). (2.83)

Again, from (2.71) and (2.72), it follows that

γ∗+pLp(a+1, c)g(z)=zLp(a+1, c)G(z)+

p+γ∗−aLp(a, c)G(z). (2.84)

Thus, by using (2.83) and (2.84) followed by (2.82), we obtain

Lp(a+1, c)f (z) Lp(a, c)g(z) −

β=(1−β)

φ(z)+ zφ(z) ar (z)+γ∗+p−a

, (2.85)

wherer (z)=Lp(a+1, c)G(z)/Lp(a, c)G(z). By usingLemma 2.20, we have

r (z)≺1+ξz (z∈ᐁ), (2.86)

whereξis given by (2.66). Letting

ar (z)+γ∗+p−a=ρeiπ θ/2 (2.87)

and using the techniques ofTheorem 2.3, the remaining part of the proof ofTheorem 2.22follows.

Remark_2.23. _{We easily ﬁnd the following:}

γ >                     

aξ−p, ifa−1

a < ξ < 2a−1

2a ,

2(a−p)−1

2 , ifξ=

2a−1 2a ,

aξ2₊_(p₊₁₋_a)ξ₋_p 1−ξ , if

2a−1

2a < ξ <1.

(2.88)

Takinga=c=pinTheorem 2.22, we get the following corollary.

Corollary_2.24. _Let

(14)

whereγ∗>max{(pξ2₊_ξ₋_p)/(₁₋_{ξ), p(ξ}₋₁₎_}_{. If}_f_∈_Ꮽ

psatisﬁes

arg

zf(z)

g(z) −β

<π2δ (0≤β < p; 0< δ≤1; z∈ᐁ) (2.90)

zg_g(z)(z)−p< pα (z∈ᐁ), (2.91)

then

arg

zF(z)

G(z) −β

<

π

2 (z∈ᐁ), (2.92)

δ=η+ 2 πtan

−1

ηsinπ /2−sin−1pξ/γ∗+p γ∗+p(1+ξ)+ηcosπ /2−sin−1pξ/γ∗+p

. (2.93)

Acknowledgment. This work was supported by Korea Research Foundation Grant KRF-2001-015-DP0013.

References

[1] S. D. Bernardi,Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969), 429–446.

[2] T. Bulboac˘a,Diﬀerential subordinations by convex functions, Mathematica (Cluj)29(52) (1987), no. 2, 105–113.

[3] B. C. Carlson and D. B. Shaﬀer,Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal.15(1984), no. 4, 737–745.

[4] P. N. Chichra,New subclasses of the class of close-to-convex functions, Proc. Amer. Math. Soc.62(1976), no. 1, 37–43.

[5] N. E. Cho, J. A. Kim, I. H. Kim, and S. H. Lee,Angular estimations of certain multivalent

functions, Math. Japon.49(1999), no. 2, 269–275.

[6] S. Fukui, J. A. Kim, and H. M. Srivastava,On certain subclass of univalent functions by

some integral operators, Math. Japon.50(1999), no. 3, 359–370.

[7] I. S. Jack,Functions starlike and convex of orderα, J. London Math. Soc. (2)3(1971), 469–474.

[8] W. Janowski,Some extremal problems for certain families of analytic functions. I, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.21(1973), 17–25.

[9] R. J. Libera,Some classes of regular univalent functions, Proc. Amer. Math. Soc.16(1965), 755–758.

[10] A. E. Livingston,p-valent close-to-convex functions, Trans. Amer. Math. Soc.115(1965), 161–179.

[11] S. S. Miller and P. T. Mocanu,Diﬀerential subordinations and inequalities in the complex

plane, J. Diﬀerential Equations67(1987), no. 2, 199–211.

[12] M. Nunokawa,On the order of strongly starlikeness of strongly convex functions, Proc. Japan Acad. Ser. A Math. Sci.69(1993), no. 7, 234–237.

[13] ,On some angular estimates of analytic functions, Math. Japon.41(1995), no. 2,

447–452.

(15)

[15] H. Saitoh and M. Nunokawa,On certain subclasses of analytic functions involving a linear

operator, S¯urikaisekikenky¯usho K¯oky¯uroku963(1996), 97–109.

[16] K. Sakaguchi,On a certain univalent mapping, J. Math. Soc. Japan11(1959), 72–75.

Nak Eun Cho: Division of Mathematical Sciences, Pukyong National University,

Pusan_608-737, Korea

E-mail address:[email protected]

J. Patel: Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar 751 004, India

E-mail address:[email protected]

G. P. Mohapatra: Silicon Institute of Technology, Near Infocity, Patia,

Bhubaneswar₇₅₁₀₃₅, India