PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 101(115) (2017), 161–168 https://doi.org/10.2298/PIM1715161A

ON A SUBCLASS OF MULTIVALENT CLOSE TO CONVEX FUNCTIONS

Muhammad Arif, Muhammad Ayaz, and Janusz Sokół

Abstract. We introduce a new subclass of multivalent close to convex functions related with Janowski functions and study some of their properties: coefficient estimates, inclusion and inverse inclusion, distortion problems and sufficiency criteria to be in these subclasses.

1. Introduction

Let A(p) denote the class of functions f(z) which are analytic and p-valent in the region 𝕌 = {z ∈ ℂ : |z| < 1} and normalized by the condition

(1.1)  f(z) = z^{p} + ∑_{k=1}^{∞} a_{k+p} z^{k+p} , (z ∈ 𝕌).

We write A(1) = A. Robertson introduced in [9] the class S^{∗}(α) of starlike functions of order α ⩽ 1, which are defined by

S^{∗}(α) := \{f ∈ A : Re( \frac{zf′(z)}{f(z)} ) > α, z ∈ 𝕌\}.

By S^{∗} = S^{∗}(0) we denote the subclasses of A which consist of univalent starlike functions. An important subclass of analytic functions is the class K of close-to-convex functions

K = \{f ∈ A : ∃β ∈ ℝ, ∃g ∈ S^{∗} : Re\{ \frac{zf′(z)}{e^{iβ} g(z)}\} > 0, z ∈ 𝕌\}.

Each close-to-convex function is univalent in the unit disc. For two functions f(z) and g(z) analytic in 𝕌, we say that f(z) is subordinate to g(z), denoted by f(z) ≺ g(z), if there is an analytic function w(z) with |w(z)| ⩽ |z| such that f(z) = g(w(z)). If g(z) is univalent, then f(z) ≺ g(z) if and only if f(0) = g(0) and f(𝕌) ⊂ g(𝕌).

2010 Mathematics Subject Classification: Primary 30C45; Secondary 30C10, 47B38.

Key words and phrases: close-to-convex functions, Janowski functions, multivalent functions, subordination, starlike functions.

Communicated by Stevan Pilipović.

In [11] Sakaguchi introduced the class S_{s}^{∗} of starlike functions with respect to symmetric points; a function f(z) ∈ A belongs to the class S_{s}^{∗}, if and only if

\frac{zf′(z)}{f(z) − f(−z)} ≺ \frac{1 + z}{1 − z} , (z ∈ 𝕌).

One can easily obtain that the function (f(z) − f(−z))/2 is starlike in 𝕌 and therefore the functions in S_{s}^{∗} are close-to-convex. Motivated from Sakaguchi’s work, Gao and Zhou [3] introduced a class K_{s}. A function f(z) ∈ A belongs to the class K_{s} if it satisfies the subordination

−\frac{z^{2}f′(z)}{g(z)g(−z)} ≺ \frac{1 + z}{1 − z} , (z ∈ 𝕌),

for some g(z) ∈ S^{∗}(1/2).

In [6] it was introduced the class K_{s}(γ) of functions satisfying

(1.2)  \frac{−z^{2}f′(z)}{g(z)g(−z)} ≺ q_{γ}(z) = \frac{1 − (1 − 2γ)z}{1 + z} .

The class K_{s}(γ) has been generalized in several directions, see the references in [7]. Recently, Xu, Srivastava and Li considered in [15] the class K_{s}(h) of functions satisfying (1.2) with a convex function h instead of q_{γ}. Şeker introduced in [12] the class K_{s}^{k}(γ), k > 1, of functions defined by (1.2) with

−z^{2−k} ∏_{ν=0}^{k−1} ε^{−ν} g(ε^{ν}z), g ∈ S^{∗}((k − 1)/k),

instead of g(z)g(−z). Moreover, Wang, Sun and Xu introduced in [14] the class MK of meromorphic functions satisfying (1.2) with γ = 1. See also the references in [14] for the other papers in this topic.

If f ∈ A(p), α < 1 and Re \frac{zf′(z)}{f(z)} > pα, z ∈ 𝕌, then we say that f is in the class S_{p}^{∗}(α) of p-valent starlike functions of order α. Using the techniques of subordination, we now introduce a subclass of p-valent analytic functions as follows.

Definition 1.1. A function f(z) ∈ A(p) is said to be in the class W_{p}(t, λ, A, B), (0 < |t| ⩽ 1, −1 ⩽ B < A ⩽ 1 and λ ∈ (0, 1]), if it satisfies

(1.3)  \frac{t^{p}z^{p+1} F′_{λ}(z)}{pg(z)g(tz)} ≺ q(z) := \frac{1 + Az}{1 + Bz}

for some g(z) ∈ S_{p}^{∗}(1/2) and F_{λ}(z) is defined by F_{λ}(z) = (1 − λ)f(z) + \frac{λ}{p}zf′(z).

In the literature, various interesting subclasses of this class have been studied from a number of different view points. For example: if we set t = −1, p = 1 and ξ = 0 in Definition 1.1, we get the class W_{1}(−1, λ, A, B) ≡ K_{s}(λ, A, B) which was studied recently by Wang and Chen [13] and further for λ = 0, A = 1 − 2γ and B = −1, we obtain the class K_{s}(γ) introduced in [6]. For more details of the related work see [1, 2, 4, 8, 12, 15].

The main object of the present paper is to introduce a subclass of p-valent analytic functions and then investigate some useful results including the coefficient estimate, sufficiency criteria to be in a class, distortion problem, radius of convexity and inclusion relationship for the new defined class.

To avoid repetition, we shall assume, unless otherwise stated, that λ ∈ (0, 1], −1 ⩽ B < A ⩽ 1, and 0 < |t| ⩽ 1.

2. Some properties of the class W_{p}(t, λ, A, B)

Theorem 2.1. Let g_{i}(z) ∈ S_{p}^{∗}(α_{i}) with α_{i} < 1. Then
(2.1)  G(z) = \frac{g_{1}(t_{1}z)g_{2}(t_{2}z)}{t_{1}^{p}t_{2}^{p}z^{p}} ∈ S_{p}^{∗}(γ),

where γ = α_{1} + α_{2} − 1 and 0 < |t_{i}| ⩽ 1, i = 1, 2.

Proof. Let g_{i}(z) ∈ S_{p}^{∗}(α_{i}). Then by definition we have

Re \frac{t_{1}zg′_{1}(t_{1}z)}{g_{1}(t_{1}z)} > pα_{1}, Re \frac{t_{2}zg′_{2}(t_{2}z)}{g_{2}(t_{2}z)} > pα_{2}, |z| < 1, 0 < |t_{i}| ⩽ 1.

By logarithmic differentiating (2.1), we obtain that

\frac{zG′(z)}{G(z)} = \frac{t_{1}zg′_{1}(t_{1}z)}{g_{1}(t_{1}z)} + \frac{t_{2}zg′_{2}(t_{2}z)}{g_{2}(t_{2}z)} − 1.

It follows that

Re \frac{zG′(z)}{G(z)} = Re \frac{t_{1}zg′_{1}(t_{1}z)}{g_{1}(t_{1}z)} + Re \frac{t_{2}zg′_{2}(t_{2}z)}{g_{2}(t_{2}z)} − 1 > pα_{1} + pα_{2} − p = pγ.

It implies that G(z) ∈ S_{p}^{∗}(γ) and it completes the proof of the theorem. □

Corollary 2.1. If g(z) ∈ S_{p}^{∗}(1/2) and 0 < |t| < 1, then

\frac{g(z)g(tz)}{t^{p}z^{p}} ∈ S_{p}^{∗}(0) := S_{p}^{∗}.

Theorem 2.2. If −1 < D, then W_{p}(t, λ, A, B) ⊂ W_{p}(t, λ, C, D). if and only if

| \frac{1 − CD}{1 − D^{2}} − \frac{1 − AB}{1 − B^{2}} | ⩽ \frac{C − D}{1 − D^{2}} − \frac{A − B}{1 − B^{2}} .

If −1 = D, then W_{p}(t, λ, A, B) ⊂ W_{p}(t, λ, C, D) if and only if

(2.2)  C ⩾ 1 − \frac{2(1 − A)}{1 − B} .

Proof. Condition (1.3) means that for z ∈ 𝕌 the values of the function

H(z) := \frac{t^{p}z^{p+1} F′_{λ}(z)}{pg(z)g(tz)}

lie in q(𝕌) because q(z) = (1 + Az)/(1 + Bz) is univalent in 𝕌. In the case B ≠ −1, q(𝕌) is a disc D(A, B), with a center S(A, B) and a radius R(A, B)

(2.3)  S(A, B) = \frac{1 − AB}{1 − B^{2}} , R(A, B) = \frac{A − B}{1 − B^{2}} ,

while it is a half-plane for B = −1. By simple computation, we can easily obtain that (1.3) is equivalent to

| \frac{t^{p}z^{p+1} F′_{λ}(z)}{pg(z)g(tz)} − \frac{1 − AB}{1 − B^{2}} | < \frac{A − B}{1 − B^{2}} , B ≠ −1,

or

(2.4)  Re\{ \frac{t^{p}z^{p+1} F′_{λ}(z)}{pg(z)g(tz)}\} > \frac{1 − A}{2} , B = −1.

Therefore, for the case B ≠ −1 D ≠ −1, the inclusion relation W_{p}(t, λ, A, B) ⊂ W_{p}(t, λ, C, D) holds when

R(A, B) ⩽ R(C, D), and |S(C, D) − S(A, B)| ⩽ R(C, D) − R(A, B).

This is equivalent to

| \frac{1 − CD}{1 − D^{2}} − \frac{1 − AB}{1 − B^{2}} | ⩽ \frac{C − D}{1 − D^{2}} − \frac{A − B}{1 − B^{2}} .

If D = −1, then by (2.4), the inclusion relation W_{p}(t, λ, A, B) ⊂ W_{p}(t, λ, C, D) holds when \frac{1−C}{2} ⩽ \frac{1−A}{1−B} . This is equivalent to (2.2). □

Lemma 2.1. If g(z) ∈ S_{p}^{∗}(1/2) and it has the form (1.1), then

|a_{n+p} t^{n} + a_{n+p−1} a_{p+1} t^{n−1} + … + a_{p+1} a_{n+p−1} t + a_{n+p} | ⩽ \frac{2p}{n} ∏_{i=1}^{n−1} ( 1 + \frac{2p}{i}) .

Proof. By virtue of Corollary 2.1, we have \frac{g(z)g(tz)}{t^{p} z^{p}} ∈ S_{p}^{∗}(0), and if

(2.5)  G(z) = \frac{g(z)g(tz)}{t^{p}z^{p}} = z^{p} + ∑_{k=p+1}^{∞} c_{k}z^{k},

then it is well known that

(2.6)  |c_{p+n} | ⩽ \frac{2p}{n} ∏_{i=1}^{n−1} ( 1 + \frac{2p}{i}) .

Substituting the series expansions of G(z) and g(z) in (2.5), we get

\frac{(z^{p} + ∑ _{k=p+1}^{∞} a_{k}z^{k})((tz)^{p} + ∑ _{k=p+1}^{∞} a_{k}(tz)^{k})}{t^{p}z^{p}} = z^{p} + ∑_{k=p+1}^{∞} c_{k}z^{k},

Comparing the coefficients of z^{n+p} , we have

(2.7)  a_{p+n} t^{n} + a_{p+1} a_{p+n−1} t^{n−1} + a_{p+2} a_{p+n−2} t^{n−2} + … + a_{p+n} = c_{p+n} .

Putting the value from (2.6) in (2.7), we get the required result. □

Theorem 2.3. Let f(z) ∈ W_{p}(t, λ, A, B) be of the form (1.1). Then

|a_{p+n} | ⩽ (A − B)[1 + ∑_{i=1}^{n−1} ( \frac{2p}{i}) ∏_{j=1}^{i−1} ( 1 + \frac{2p}{j}) ] + 1 + \frac{2p}{n} ∏_{i=1}^{n−1} ( \frac{2p}{i}) .

Proof. Let f(z) ∈ W_{p}(t, λ, A, B). Then we have
(2.8)  \frac{zF′_{λ}(z)}{pG(z)} ≺ \frac{1 + Az}{1 + Bz} ,

where G(z) is given by (2.5). If we put

(2.9)  q(z) = \frac{zF′_{λ}(z)}{pG(z)},

it follows from (2.8) that

q(z) = 1 + ∑_{n=1}^{∞} q_{n}z^{n} ≺ \frac{1 + Az}{1 + Bz} = 1 + ∑_{n=1}^{∞} A_{n}z^{n}, A_{n} = (A − B)(−B)^{n−1} .

The function (1 + Az)/(1 + Bz) is convex univalent, hence applying the well known Rogosinski result [10], we obtain

(2.10)  |q_{n}| ⩽ A_{1} = A − B, n = 1, 2, . . ..

Now by putting the series expansions of f(z), G(z) and q(z) in (2.9) and then comparing the coefficients of z^{n+p} , we obtain

\frac{1}{p}(p + nλ)(p + n)a_{p+n} = c_{p+n} + q_{p+1} c_{p+n−1} + ⋯+ q_{p+n−1} c_{p+1} + q_{p+n} .

By using (2.10) and (2.6) we have

\frac{1}{p}(p + nλ)(p + n)|a_{p+n} | ⩽ (A − B)(|c_{p+n−1} | + ⋯+ |c_{p+1} | + 1) + |c_{p+n} |

⩽ (A − B)[1 + ∑_{i=1}^{n−1} |c_{p+i} |] + ( \frac{2p}{n}) ∏_{i=1}^{n−1} ( 1 + \frac{2p}{i})

⩽ (A − B)[1 + ∑_{i=1}^{n−1} ( \frac{2p}{i}) ∏_{j=1}^{i−1} ( 1 + \frac{2p}{j}) ] + ( \frac{2p}{n}) ∏_{i=1}^{n−1} ( 1 + \frac{2p}{i}) .

This completes the proof. □

Theorem 2.4. If g(z) ∈ S_{p}^{∗}(1/2) and f(z) ∈ A(p) is of the form (1.1), and if it satisfies the condition

(2.11)  \frac{1 + A}{2} − ∑_{n=1}^{∞} \frac{(1 + nλ/p)(p + n)}{p} |a_{p+n} | − \frac{1 − A}{2} ∑_{n=1}^{∞} |c_{p+n} | > 0,

where c_{p+n} is given by

G(z) = \frac{g(z)g(tz)}{t^{p}z^{p}} = z^{p} + ∑_{n=1}^{∞} c_{p+n} z^{p+n} ,

then f(z) ∈ W_{p}(t, λ, A, −1). Moreover, if it satisfies the condition

(2.12) \frac{A − B}{1 + B} −∑_{n=1}^{∞} \{ \frac{(1 + nλ/p)(p + n)}{p} |a_{p+n} |+( \frac{A − B}{1 − B^{2}} + \frac{1 − AB}{1 − B^{2}} ) |c_{p+n} |\} > 0,

then f(z) ∈ W_{p}(t, λ, A, B), B > −1.

166 ARIF, AYAZ, AND SOKÓŁ

Proof. To prove that f(z) ∈ W_{p}(t, λ, A, −1), it is enough to show that

| \frac{zF′_{λ}(z)}{pG(z)} | > \frac{1 − A}{2} , z ∈ 𝕌

or equivalently to show that

| \frac{F′_{λ}(z)}{pz^{p−1}} | − | \frac{(1 − A)G(z)}{2z^{p}} | > 0, z ∈ 𝕌.

We have

| \frac{F′_{λ}(z)}{pz^{p−1}} | − | \frac{(1 − A)G(z)}{2z^{p}} |

= |1 + ∑_{n=1}^{∞} \frac{(1 + nλ/p)(p + n)}{p} a_{p+n} z^{n} | − | \frac{(1 − A)}{2} (1 + ∑_{n=1}^{∞} c_{p+n} z^{n})|

⩾ 1 − | ∑_{n=1}^{∞} \frac{(1 + nλ/p)(p + n)}{p} a_{p+n} z^{n} | − | \frac{(1 − A)}{2} (1 + ∑_{n=1}^{∞} c_{p+n} z^{n})|

⩾ 1 − ∑_{n=1}^{∞} \frac{(1 + nλ/p)(p + n)}{p} |a_{p+n} | − \frac{(1 − A)}{2} − \frac{(1 − A)}{2} ∑_{n=1}^{∞} |c_{p+n} |

= \frac{1 + A}{2} − ∑_{n=1}^{∞} \frac{(1 + nλ/p)(p + n)}{p} |a_{p+n} | − \frac{1 − A}{2} ∑_{n=1}^{∞} |c_{p+n} | ⩾ 0

by (2.11). For the case B > −1 it suffices to show that

|z \frac{F′_{λ}(z)}{pG(z)} − S(A, B)| < R(A, B),

where S(A, B) and R(A, B) are given in (2.3). This is equivalent to

| \frac{F′_{λ}(z)}{pz^{p−1}} − \frac{G(z)S(A, B)}{z^{p}} | < | \frac{G(z)R(A, B)}{z^{p}} |, z ∈ 𝕌.

We have

| \frac{G(z)R(A, B)}{z^{p}} | − | \frac{F′_{λ}(z)}{pz^{p−1}} − \frac{G(z)S(A, B)}{z^{p}} |

= R(A, B)|1 + ∑_{n=1}^{∞} c_{p+n} z^{n} |

− |1 + ∑_{n=1}^{∞} \frac{(1 + nλ/p)(p + n)}{p} a_{p+n} z^{n} − S(A, B)1 + ∑_{n=1}^{∞} c_{p+n} z^{n} |

= R(A, B)|1 + ∑_{n=1}^{∞} c_{p+n} z^{n} |

− | \frac{B(A − B)}{1 − B^{2}} + ∑_{n=1}^{∞} \{ \frac{(1 + nλ/p)(p + n)}{p} a_{p+n} − S(A, B)c_{k}\}z^{n} |

ON MULTIVALENT CLOSE TO CONVEX FUNCTIONS 167

⩾ R(A, B) − R(A, B) ∑_{n=1}^{∞} |c_{p+n} |

− \frac{B(A − B)}{1 − B^{2}} − ∑_{n=1}^{∞} \{ \frac{(1 + nλ/p)(p + n)}{p} |a_{p+n} | + S(A, B)|c_{p+n} |\}

= \frac{A − B}{1 + B} − ∑_{n=1}^{∞} \{ \frac{(1 + nλ/p)(p + n)}{p} |a_{p+n} | + (R(A, B) + S(A, B))|c_{p+n} |\} > 0

by (2.12). □

Theorem 2.5. Let f(z) ∈ W_{p}(t, 0, A, B). Then

\frac{1 − Ar}{1 − Br} \frac{(1 − r)^{2p}p}{r^{p+1}} ⩽ |f′(z)| ⩽ \frac{1 + Ar}{1 + Br} \frac{(1 + r)^{2p}p}{r^{p+1}} .

Proof. Suppose that f(z) ∈ W_{p}(t, 0, A, B). Then by using definition of subordination between analytic functions, we can write

(2.13)  \frac{1 − Ar}{1 − Br} ⩽ \frac{1 − A|w(z)|}{1 − B|w(z)|} ⩽ | \frac{zf′(z)}{pG(z)} | ⩽ \frac{1 + A|w(z)|}{1 + B|w(z)|} ⩽ \frac{1 + Ar}{1 + Br} ,

where w(z) is the Schwarz function with w(0) = 0, |w(z)| < |z| = r. Now it is shown in Corollary 2.1 that G(z) = \frac{g(z)g(tz)}{t^{p} z^{p}} ∈ S_{p}^{∗}(0), thus we have

(2.14)  \frac{r^{p}}{(1 + r)^{2p}} ⩽ |G(z)| ⩽ \frac{r^{p}}{(1 − r)^{2p}} .

Now by using (2.14) in (2.13), we obtain the required result. □

Theorem 2.6. Let f(z) ∈ W_{p}(t, 0, 1, B). Then

1 + \frac{zf′′(z)}{f′(z)} > 0, |z| < r_{0},

where r_{0} is the smallest positive root of the equation

(2.15)  p(1 − r)^{2}(1 − Br) − r(1 + r)(1 − B) = 0.
Proof. Suppose f(z) ∈ W_{p}(t, 0, 1, B) and let

\frac{zf′(z)}{pG(z)} = q(w(z)) := s(z), |w(z)| ⩽ |z|.

where q(z) is given in (1.3). Logarithmic differentiation gives us

(2.16)  1 + \frac{zf′′(z)}{f′(z)} = \frac{zG′(z)}{G(z)} + \frac{zs′(z)}{s(z)} .

Applying [5, Theorem 3] we have

(2.17)  Re \frac{zs′(z)}{s(z)} > −\frac{(1 − B)r}{(1 − r)(1 − Br)} , |z| < r.

Moreover, G(z) ∈ S_{p}^{∗}(0), so

(2.18)  Re \frac{zG′(z)}{pG(z)} > \frac{1 − r^{p}}{1 + r^{p}} , |z| < r.

Applying (2.17) and (2.18) in (2.16), we obtain

Re\{1 + \frac{zf′′(z)}{f′(z)} \} > p\frac{1 − r^{p}}{1 + r^{p}} − \frac{(1 − B)r}{(1 − r)(1 − Br)} > p\frac{1 − r}{1 + r} − \frac{(1 − B)r}{(1 − r)(1 − Br)}

= \frac{p(1 − r)^{2}(1 − Br) − r(1 + r)(1 − B)}{(1 − r^{2})(1 − Br)}

and this is positive for r < r_{0}, where r_{0} is described in (2.15). □

References

1. I. B. Bapana, S. Nandwana, Coefficient, distortion and growth inequalities for certain p-valent close to convex functions, J. Appl. Math. Stat. Inform. 9(2) (2013), 57–62.

2. N. E. Cho, O. S. Kown, V. Ravichandran, Coefficient, distortion and growth inequalities for certain close to convex functions, J. Inequal. Appl. 2011 (2011), Article ID 100, 7 pp.

3. C. Y. Gao, S. Q. Zhou, On a class of analytic functions related to the starlike functions, Kyungpook Math. J. 45 (2005), 123–130.

4. S. P. Goyal, P. Goswami, On certain properties for a subclass of close to convex functions, J. Class. Anal. 1(2) (2012), 103–112.

5. W. Janowski, Some extremal problems for certain families of analytic functions, Ann. Pol. Math. 28 (1973) 297–326.

6. J. Kowalczyk, E. Leś-Bomba, On a subclass of close-to-convex functions, Appl. Math. Lett. 23 (2010), 1147–1151.

7. J. Kowalczyk, E. Leś, J. Sokół, Radius problems in a certain subclass of close-to-convex functions, Houston J. Math. 40(4) (2014), 1061–1072.

8. R. K. Raina, H. M. Srivastava, A certain subclass of analytic functions associated with operators of fractional calculus, Comput. Math. Appl. 32 (1996), 13–19.

9. M. S. Robertson, Certain classes of starlike functions, Mich. Math. J. 76(1) (1954), 755–758.

10. W. Rogosinski, On the coefficients of subordinate functions, Proc. Lond. Math. Soc. 48 (1943), 48–82.

11. K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan 11 (1959), 72–75.

12. B. Şeker, On certain new subclass of close-to-convex functions, Appl. Math. Comput. 218 (2012), 1041–1045.

13. Z. G. Wang, D. Z. Chen, On a subclass of close to convex functions, Hacet. J. Math. Stat. 38(2) (2009), 95–101.

14. Z. G. Wang, Y. Sun, N. Xu Some properties of certain meromorphic close-to-convex functions, Appl. Math. Lett. 25 (2012), 454–460.

15. Q. H. Xu, H. M. Srivastava, Z. Li, A certain subclass of analytic and close-to-convex functions, Appl. Math. Lett. 24 (2011), 396–401.

Department of Mathematics (Received 06 06 2015) Abdul Wali Khan University Mardan Khyber Pakhtunkhwa, Pakistan

 marifmaths@hotmail.com

Faculty of Mathematics and Natural Sciences, University of Rzeszów Rzeszów Poland

 jsokol@prz.edu.pl