The main purpose of this paper is to determine the conditions of starlikeness for certain class of multivalent analytic functions. Relevant connections of the results presented here with those obtained in earlier works are pointed out.

1. Introduction and Main Result

Let 𝒜p denote the class of functions of the form
(1)f(z)=zp+∑n=p+1∞anzn(p∈ℕ:={1,2,…}),
which are analytic in the open unit disk 𝕌:={z:z∈ℂand|z|<1}. For convenience, we set 𝒜1=:𝒜. A function f∈𝒜p is said to be in the class 𝒮p*(ϱ) of p-valent starlike functions of order ϱ in 𝕌, if it satisfies the following inequality:
(2)ℜ(zf′(z)f(z))>ϱ(0≦ϱ<p;z∈𝕌).
For simplicity, we write 𝒮1*(0)=:𝒮*.

In [1], Chichra introduced the class ℛ of analytic functions f∈𝒜 which satisfy the condition
(3)ℜ(f′(z)+zf′′(z))>0(z∈𝕌).
He proved that the members of ℛ are univalent in 𝕌. Later, R. Singh and S. Singh [2] showed that ℛ⊂𝒮*. Recently, Gao and Zhou [3] considered the subclass ℛ(β,γ) of 𝒜 which is defined by
(4)ℛ(β,γ)≔{f∈𝒜:ℜ(f′(z)+βzf′′(z))>γ}hhhhhhhhhhhhhhhhhhh(β>0;γ<1;z∈𝕌).
They derived some mapping properties of this class. Moreover, several authors discussed some related analytic function classes associated with the class ℛ (see [4–7]). By using the method of differential subordination, Yang and Liu [8] generalized the above works and studied the subclass 𝒯p(A,B,γ,α) of 𝒜p which satisfies the condition
(5)f′(z)+αzf′′(z)≺h(z)(z∈𝕌),
where
(6)h(z)={(1+Az1+Bz)γ,(A≦1;0<γ<1),1+Az1+Bz,(γ=1).
In [9], Owa et al. introduced a new subclass ℛp(α,β,γ;j) of 𝒜p which satisfies the inequality
(7)ℜ(αf(j)(z)zp-j+βf(j+1)(z)zp-j-1)>γ(j∈{0,1,2,…,p};p∈ℕ;z∈𝕌),
where α>0, β>0, and (throughout this paper unless otherwise mentioned) the parameters γ and δ are constrained as follows:
(8)γ<δ:=p![α+(p-j)β](p-j)!(α>0;β>0;p∈ℕ;j∈{0,1,2,…,p}).
The extreme points, coefficient inequalities, radius of starlikeness, and inclusion relationship for the class ℛp(α,β,γ;j) are derived. By setting p=j=α=1, it is easy to see that the class ℛp(α,β,γ;j) reduces to the class ℛ(β,γ). If we set p=j=α=γ=1 in the class ℛp(α,β,γ;j), then it reduces to the class ℛ(γ), which was studied earlier by Silverman [10], R. Singh and S. Singh [2, 11], independently.

For some recent investigations on the starlikeness of analytic functions, one can refer to [12–20]. In the present paper, we aim at deriving the conditions of starlikeness for the class ℛp(α,β,γ;j). The main result is presented below.

Theorem 1.

Let β≧α>0. Then

ℛp(α,β,γ;1)⊂𝒮p* for γ1≦γ<p[α+(p-1)β], where γ1 is the solution of the following equation:
(9)-p2(∑n=2∞(-1)n-1(n+p-1)[α+(n+p-2)β]1+2{p[α+(p-1)β]-γ1}×∑n=2∞(-1)n-1(n+p-1)[α+(n+p-2)β])=γ1β+β-αβ×(∑n=2∞(-1)n-1α+(n+p-2)βp+2{p[α+(p-1)β]-γ1}×∑n=2∞(-1)n-1α+(n+p-2)β);

ℛp(α,β,γ;j)⊂𝒮p*(j-1)(j∈{2,3,…,p}) for γ2≦γ<p[α+(p-1)β], where γ2 is the solution of the following equation:
(10)-p-j+12(∑n=2∞(-1)n-1α+(n+p-j)βp!(p-j+1)!+2(δ-γ2)×∑n=2∞(-1)n-1α+(n+p-j)β)=γ2β+β-αβ×(∑n=2∞(-1)n-1α+(n+p-j)βp!(p-j)!+2(δ-γ2)×∑n=2∞(-1)n-1α+(n+p-j-1)β).

2. Preliminary Results

In order to establish our main theorem, we will require the following lemmas.

Lemma 2 (see [<xref ref-type="bibr" rid="B10">9</xref>]).

A function f∈ℛp(α,β,γ;j) if and only if f can be expressed as follows:
(11)f(z)=zp+2(δ-γ)×∫|x|=1(∑n=p+1∞(n-j)!n![α+(n-j)β]xn-pzn)dμ(x),
where μ(x) is the probability measure on 𝕏:={x∈ℂ:|x|=1}.

The proof of the following lemma is much akin to that of Theorem 1 which was obtained by Nunokawa et al. [21] (see also Liu [22] and Yang [23]). We, therefore, choose to omit the analogous details involved.

Lemma 3.

If f∈𝒜p satisfies the inequality
(12)ℜ(zf(j)(z)f(j-1)(z))>0(j∈{1,2,…,p};z∈𝕌),
then f∈𝒮p*(j-1).

Let ϕ be a nonconstant regular function in 𝕌. If |ϕ| attains its maximum value on the circle |z|=r<1 at z0, then
(13)z0ϕ′(z0)=kϕ(z0),
where k≧1 is a real number.

We now give the lower bounds of the following continuous linear operators:
(14)ℒ1(f)=ℜ(f(z)zp)(f∈ℛp(α,β,γ;1);z∈𝕌),ℒ2(f)=ℜ(f(j)(z)zp-j)(f∈ℛp(α,β,γ;j);hhhhhhhhhhhhhhhhhhhj∈{1,2,…,p};z∈𝕌f∈ℛp(α,β,γ;j))
acting on the class ℛp(α,β,γ;j), which played crucial role in the proof of our main result.

Lemma 5.

If f∈ℛp(α,β,γ;j), then, for |z|≦r<1, one has the following.

When j∈{1,2,…,p}, then
(15)ℜ(f(j)(z)zp-j)≧p!(p-j)!+2(δ-γ)∑n=2∞(-r)n-1α+(n+p-j-1)β>p!(p-j)!+2(δ-γ)∑n=2∞(-1)n-1α+(n+p-j-1)β.

This inequality is sharp.

When j=1, then
(16)ℜ(f(z)zp)≧1+2(δ1-γ)∑n=2∞(-r)n-1(n+p-1)[α+(n+p-2)β]>1+2(δ1-γ)∑n=2∞(-1)n-1(n+p-1)[α+(n+p-2)β],

where δ1=p[α+(p-1)β]. The inequality is sharp.

Proof.

By Lemma 2, we know that
(17)f(z)=zp+2(δ-γ)∑n=p+1∞(n-j)!n![α+(n-j)β]zn
is the extreme function of the class ℛp(α,β,γ;j). Thus, we only need to consider the function f defined by (17); it follows that
(18)f(j)(z)zp-j=p!(p-j)!+2(δ-γ)∑n=2∞(-r)n-1α+(n+p-j-1)βzn-1.
We note that (18) can be written as follows:
(19)f(j)(z)zp-j=p!(p-j)!+2(δ-γ)β∫01t(α/β)+p-jz1-tzdt.
Thus, we find from (19) that
(20)ℜ(f(j)(z)zp-j)=p!(p-j)!+2(δ-γ)β×∫01t(α/β)+p-jℜ(z1-tz)dt.
Moreover, we observe that the function
(21)k(z)=z1-tz(0≦t≦1)
is convex in 𝕌, k(z¯)=k(z)¯, and k(z) maps real axis to real axis; we have
(22)-r1+tr≦ℜ(z1-tz)≦r1-tr(|z|≦r<1).
Upon substituting (22) into (20) and expanding the integrand into the power series of t and integrating it, we can easily get (15). The sharpness of (15) can be seen from (18).

By similarly applying the method of proof of (15), we also can prove (16) holds true. The sharpness of (16) can be found in (17).

3. Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>Proof.

Suppose that f∈ℛp(α,β,γ;j) with β≧α>0. It follows from (7) that
(23)ℜ(f(j)(z)zp-j+βαf(j+1)(z)zp-j-1)>γα(z∈𝕌).
By noting that
(24)βα(f(j)(z)zp-j+f(j+1)(z)zp-j-1)=f(j)(z)zp-j+βαf(j+1)(z)zp-j-1+(βα-1)f(j)(z)zp-j,
we get
(25)ℜ(f(j)(z)zp-j+f(j+1)(z)zp-j-1)=αβℜ(f(j)(z)zp-j+βαf(j+1)(z)zp-j-1)+β-αβℜ(f(j)(z)zp-j).
Thus, we can easily find from (15), (23), and (25) that
(26)ℜ(f(j)(z)zp-j+f(j+1)(z)zp-j-1)>γβ+β-αβ×(p!(p-j)!+2(δ-γ)∑n=2∞(-1)n-1α+(n+p-j-1)β).
We now set
(27)f(j)(z)f(j-1)(z)=(p-j+1)1+ω(z)1-ω(z)hhhhh(j∈{1,2,…,p};z∈𝕌).
Then ω is analytic in 𝕌 with ω(0)=0. It follows from (27) that
(28)f(j)(z)zp-j+f(j+1)(z)zp-j-1=(p-j+1)f(j-1)(z)zp-j+1[(1+ω(z)1-ω(z))2+2zω′(z)(1-ω(z))2].
At the same time, we can claim that |ω(z)|<1. Indeed, if not, there exists a point z0∈𝕌 such that
(29)max|z|≦|z0||ω(z)|=|ω(z0)|=1;
by Lemma 4, we obtain
(30)z0ω′(z0)=kω(z0)=keiθ(0<θ<2π;k≧1).
For z=z0, by virtue of (28), we split it into two cases to prove the following.

When j=1, in view of (16), we get
(31)ℜ(f′(z0)z0p-1+f′′(z0)z0p-2)=pℜ(f(z0)z0p[(1+eiθ1-eiθ)2+2keiθ(1-eiθ)2])≦-pk2sin2(θ/2)ℜ(f(z0)z0p)≦-p2ℜ(f(z0)z0p)≦-p2((-1)n-1(n+p-1)[α+(n+p-2)β]∑n=2∞1+2{p[α+(p-1)β]-γ}×∑n=2∞(-1)n-1(n+p-1)[α+(n+p-2)β]).

Let β≧α>0. If γ satisfies the condition
(32)-p2((-1)n-1(n+p-1)[α+(n+p-2)β]∑n=2∞1+2{p[α+(p-1)β]-γ}×∑n=2∞(-1)n-1(n+p-1)[α+(n+p-2)β])≦γβ+β-αβ×((-1)n-1α+(n+p-j-1)β∑n=2∞p+2{p[α+(p-1)β]-γ}×∑n=2∞(-1)n-1α+(n+p-j-1)β),

we have a contradiction to (26) at z=z0; the smallest γ satisfies (32) is solution γ1 of (9). This implies that if β≧α>0 and γ1≦γ<p[α+(p-1)β], we have |ω(z)|<1. Thus, we conclude that f∈𝒮p*.

When j∈{2,3,…,p}, by means of (15), we have
(33)ℜ(f(j)(z0)z0p-j+f(j+1)(z0)z0p-j-1)=(p-j+1)ℜ(f(j-1)(z0)z0p-j+1[(1+eiθ1-eiθ)2+2keiθ(1-eiθ)2])≦-(p-j+1)k2sin2(θ/2)ℜ(f(j-1)(z0)z0p-j+1)≦-p-j+12ℜ(f(j-1)(z0)z0p-j+1)≦-p-j+12×(p!(p-j+1)!+2(δ-γ)∑n=2∞(-1)n-1α+(n+p-j)β).

Let β≧α>0. If γ satisfies the following inequality,
(34)-p-j+12×(p!(p-j+1)!+2(δ-γ)∑n=2∞(-1)n-1α+(n+p-j)β)≦γβ+β-αβ×(p!(p-j)!+2(δ-γ)∑n=2∞(-1)n-1α+(n+p-j-1)β),

we have a contradiction to (26) at z=z0; the smallest γ satisfies (34) is solution γ2 of (10). This shows that if β≧α>0 and γ2≦γ<δ, we have |ω(z)|<1. It follows from (27) that
(35)ℜ(zf(j)(z)f(j-1)(z))>0(j∈{2,3,…,p};z∈𝕌).

Therefore, by Lemma 3, we deduce that
(36)ℛp(α,β,γ;j)∈𝒮p*(j-1)(j∈{2,3,…,p}).
The proof of the theorem is thus completed.Remark 6.

If β<α (α>0;β>0), we cannot find the number γ(α,β) such that
(37)ℛp(α,β,γ;j)∈𝒮p*(j-1)(j∈{2,3,…,p}),
since the continuous linear operators ℒ1(f) and ℒ2(f) acting on ℛp(α,β,γ;j) do not exist in sharp upper bounds.

Putting p=α=1 in the first part of Theorem 1, we can get the following result.

Corollary 7.

Let β≧1. Then ℛ(β,γ)⊂𝒮* for γ3≦γ<1, where γ3 is the solution of the following equation:
(38)1-32β=γ3+(1-γ3)∑n=2∞(-1)n-1[β+2n(β-1)]n[1+(n-1)β].

Remark 8.

Corollary 7 corrects some errors of Theorem 4 in [3].

By setting p=α=β=1 in the first part of Theorem 1, we also can get the following criterion for starlikeness obtained by Silverman [10].

Corollary 9.

Consider ℛ(γ)⊂𝒮* for (6-π2)/(24-π2)≦γ<1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grant nos. 11301008, 11226088, and 11101053, the Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province under Grant no. 2013GGJS-146, and the Natural Science Foundation of Educational Committee of Henan Province under Grant 14B110012 of China. The authors are grateful to the referees for their valuable comments and suggestions which essentially improved the quality of this paper.

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