International Journal of Analysis and Applications
ISSN 2291-8639
Volume 12, Number 1 (2016), 62-65
http://www.etamaths.com

ON p-VALENTLY MEROMORPHIC-STRONGLY STARLIKE AND CONVEX

FUNCTIONS

RAHIM KARGAR1,∗, ALI EBADIAN2 AND JANUSZ SOKÓ L3

Abstract. In this paper, we obtain sufficient conditions for analytic function f(z) in the punctured

unit disk to be p-valently meromorphic-strongly starlike and p-valently meromorphic-strongly convex
of order β and type α. Some interesting corollaries of the results presented here are also discussed.

1. Introduction

Let H be the class of functions that are analytic in the unit disk U and let A be the class of functions
of the form:

f(z) = z + a2z
2 + a3z

3 + · · · ,
which are analytic in U.

Let Σ(p) denote the class of meromorphically p-valent functions f(z) of the form

f(z) = z−p +

∞∑
k=1

ak−pz
k−p p ∈ N := {1, 2, 3, . . .},

which are analytic in the punctured unit disk U∗ = {z ∈ C : 0 < |z| < 1} = U\{0}. Further, we write
that Σ(1) = Σ. For a function f ∈ Σ(p), we say that it is p-valently meromorphic-strongly starlike of
order 0 < β ≤ 1 and type α (0 ≤ α < p) if

(1)

∣∣∣∣arg
(
−
zf′(z)

f(z)
−α

)∣∣∣∣ < πβ2 z ∈ U.
The corresponding class is denoted by ST Σ(α,β). We note that ST Σ(α, 1), is the class of p-valently
meromorphic starlike functions of order α (see [6]) and ST Σ(0,β) is the class of p-valently meromorphic-
strongly starlike functions of order β. Furthermore, a function f ∈ Σ(p) is said to be in the class
SKΣ(α,β) of p-valently meromorphic-strongly convex of order β and type α if and only if

(2)

∣∣∣∣arg
(
−1 −

zf′′(z)

f′(z)
−α

)∣∣∣∣ < πβ2 z ∈ U,
for some real 0 < β ≤ 1 and 0 ≤ α < p. In particular, SKΣ(α, 1), is the class of p-valently meromorphic
convex functions of order α (cf. [6]) and SKΣ(0,β) is the class of p-valently meromorphic-strongly
convex functions of order β.

A function f ∈A is said to be in the class ST (β) of strongly starlike function of order β, 0 ≤ β < 1,
if it satisfies the inequality ∣∣∣∣arg

(
zf′(z)

f(z)

)∣∣∣∣ < πβ2 z ∈ U.
A function f(z) belonging to the class SK(β) is said to be strongly convex of order β in U if and only
if zf′(z) ∈ST (β) (see [2, 3]).

For proving our results we need the following Lemma.

2010 Mathematics Subject Classification. 30C45, 30C55.

Key words and phrases. analytic; meromorphic; p-valent; strongly starlike; strongly convex.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

62



ON p-VALENTLY MEROMORPHIC FUNCTIONS 63

Lemma 1. (see [1], [5]). Let b(z) ∈ H be continuous on U, b(0) = 0, supz∈U |b(z)| = 1 and c =
supz∈U

∫ 1
0
|b(tz)|dt. For 0 < β ≤ 1 let

λ(β) =
sin(πβ/2)√

1 + 2c cos(πβ/2) + c2
.

If f ∈A and
|f′(z) − 1| ≤ λ(β)|b(z)| z ∈ U,

then f is strongly starlike of order β. Additionally, if

b(t) = max
0≤ϕ≤2π

|b(teiϕ)| 0 ≤ t ≤ 1,

then the constant λ(β) cannot be replaced by any larger number without violating the conclusion.

The Lemma 1, without the sharpness part, was previously obtained by Ponnusamy and Singh in
[4].

In this work, we obtain some sufficient conditions for p-valently meromorphic functions.

2. Main Results

Our first result is contained in the following.

Theorem 2. Assume that f(z) 6= 0 for z ∈ U∗. If f ∈ Σ(p) satisfies

(3)

∣∣∣∣∣
(
f(z)

z−α

) 1
α−p

(
f′(z)

f(z)
+
α

z

)
+ p−α

∣∣∣∣∣ < (p−α)λ(β)|b(z)| z ∈ U∗,
then f is p-valently meromorphic-strongly starlike of order β and type α.

Proof. Assume that f ∈ Σ(p). Let us define the function g(z) by

(4) g(z) =

(
f(z)

z−α

) 1
α−p

= z + · · · z ∈ U∗.

Then g(z) ∈A and

|g′(z) − 1| =
1

p−α

∣∣∣∣∣
(
f(z)

z−α

) 1
α−p

(
f′(z)

f(z)
+
α

z

)
+ p−α

∣∣∣∣∣ .
Now, by means of the condition of the theorem and applying Lemma 1 we find that g(z) is strongly
starlike function of order β. Note that from (4) we have

zg′(z)

g(z)
=

1

p−α

(
−
zf′(z)

f(z)
−α

)
.

Since g(z) is strongly starlike of order β, thus∣∣∣∣arg
{

1

p−α

(
−
zf′(z)

f(z)
−α

)}∣∣∣∣ < πβ2 .
This shows that the proof is completed. �

Putting α = 0 in Theorem 2, we have:

Corollary 3. Assume that f(z) 6= 0 for z ∈ U∗. If f ∈ Σ(p) satisfies∣∣∣∣∣ 1p√f(z)
(
f′(z)

f(z)

)
+ p

∣∣∣∣∣ < pλ(β)|b(z)| z ∈ U∗,
then f is p-valently meromorphic-strongly starlike of order β.

Setting b(z) = z and p = β = 1 in Theorem 2, we obtain the following result:



64 KARGAR, EBADIAN AND SOKÓ L

Corollary 4. Assume that f(z) 6= 0 for z ∈ U∗. If f ∈ Σ satisfies∣∣∣∣∣
(
f(z)

z−α

) 1
α−1

(
f′(z)

f(z)
+
α

z

)
+ 1 −α

∣∣∣∣∣ < 2√5 (1 −α) z ∈ U∗,
then f is meromorphic starlike function of order α.

If we take α = 0 in Corollary 4, we obtain the following result:

Corollary 5. Assume that f(z) 6= 0 for z ∈ U∗. If f ∈ Σ satisfies∣∣∣∣∣
(

1

f(z)

)2
f′(z) + 1

∣∣∣∣∣ < 2√5 ≈ 0.894427... z ∈ U∗,
then f is meromorphic starlike functions.

Next we derive the following.

Theorem 6. Assume that f′(z) 6= 0 for z ∈ U∗. If f ∈ Σ(p) satisfies

(5)

∣∣∣∣∣
(

f′(z)

−pz−p−1

) 1
α−p

(
1 +

zf′′(z)

f′(z)
+ p

)∣∣∣∣∣ < (p−α)λ(β)|b(z)| z ∈ U∗,
then f is p-valently meromorphic-strongly convex of order β and type α.

Proof. Let f ∈ Σ(p) and define the function p(z) by

(6) p(z) =

∫ z
0

(
f′(t)

−pt−p−1

) 1
α−p

dt = z + · · · z ∈ U∗.

Further, let

(7) h(z) = zp′(z) = z

(
f′(z)

−pz−p−1

) 1
α−p

= z + · · · z ∈ U∗.

We see that p(z) and h(z) belongs to A. Differentiating from (7), we have

h′(z) =
1

α−p

(
f′(z)

−pz−p−1

) 1
α−p

(
1 +

zf′′(z)

f′(z)
+ α

)
.

Further we have

|h′(z) − 1| =
1

p−α

∣∣∣∣∣
(

f′(z)

−pz−p−1

) 1
α−p

(
1 +

zf′′(z)

f′(z)
+ p

)
− (α−p)

∣∣∣∣∣ < λ(β)|b(z)|.
Therefore, applying of the Lemma 1 gives us that

h(z) = zp′(z) ∈ST (β) ⇒ p(z) ∈SK(β).
Since

zp′′(z)

p′(z)
=

1

α−p

(
zf′′(z)

f′(z)
+ 1 + p

)
,

therefore ∣∣∣∣arg
(

1 +
zp′′(z)

p′(z)

)∣∣∣∣ =
∣∣∣∣arg 1p−α

(
−1 −

zf′′(z)

f′(z)
−α

)∣∣∣∣ < πβ2 ,
which imply that f(z) is p-valently meromorphic-strongly convex of order β and type α. This completes
the proof. �

Putting α = 0 in Theorem 6, we have:

Corollary 7. Assume that f′(z) 6= 0 for z ∈ U∗. If f ∈ Σ(p) satisfies∣∣∣∣∣ p
√
−pz−p−1
f′(z)

(
1 +

zf′′(z)

f′(z)
+ p

)∣∣∣∣∣ < pλ(β)|b(z)| z ∈ U∗,
then f is p-valently meromorphic-strongly convex of order β.



ON p-VALENTLY MEROMORPHIC FUNCTIONS 65

Setting b(z) = z and p = β = 1 in Theorem 6, we obtain the following result:

Corollary 8. Assume that f′(z) 6= 0 for z ∈ U∗. If f ∈ Σ satisfies∣∣∣∣(−z2f′(z)) 1α−1
(

2 +
zf′′(z)

f′(z)

)∣∣∣∣ < 2√5 (1 −α) z ∈ U∗,
then f is meromorphic convex function of order α.

If we take α = 0 in Corollary 8, we obtain the following result:

Corollary 9. Assume that f′(z) 6= 0 for z ∈ U∗. If f ∈ Σ satisfies∣∣∣∣ −1z2f′(z)
(

2 +
zf′′(z)

f′(z)

)∣∣∣∣ < 2√5 ≈ 0.894427 . . . z ∈ U∗,
then f is convex meromorphic function.

References

[1] R. Kargar and R. Aghalary, Sufficient condition for strongly starlike and convex functions, Serdica Math. J. 40
(2014), 13-18.

[2] P. T. Mocanu, Alpha-convex integral operators and strongly starlike functions, Stud. Univ. Babes-Bolyai Math. 34

(1989), 18-24.
[3] M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. Japan Acad. Ser. A Math.

Sci. 69(7) (1993), 234-237.

[4] S. Ponnusamy and V. Singh, Criteria for strongly starlike functions, Complex Variables, Theory Appl. 34(3) (1997),
267-291.

[5] F. Rønning, S. Ruscheweyeh and S. Samaris, Sharp starlikeness conditions for analytic functions with bounded

derivative, J. Aust. Math. Soc. Ser. A, 69 (2000), 303-315.
[6] M. San and H. Irmak, Ordinary differential operator and some of its applications to certain meromorphically p-valent

functions, App. Math. Comput. 218 (2011), 817-821.

1Young Researchers and Elite Club, Urmia Branch, Islamic Azad University, Urmia, Iran

2Department of Mathematics, Payame Noor University, P.O. Box 19395-3697 Tehran, Iran

3Department of Mathematics, Institute of Mathematics, University of Rzeszów, ul. Rejtana 16A, 35-310
Rzeszów, Poland

∗Corresponding author: rkargar1983@gmail.com