pISSN 1225-6951 eISSN 0454-8124 c
Kyungpook Mathematical Journal
Subclasses of Starlike and Convex Functions Associated with
Pascal Distribution Series
Basem Aref Frasin∗
Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq, Jordan
e-mail : bafrasin@yahoo.com Sondekola Rudra Swamy
Department of Computer Science and Engineering, RV College of Engineering, Ben-galuru - 560 059, Karnataka, India
e-mail : mailtoswamy@rediffmail.com Abbas Kareem Wanas
Department of Mathematics, College of Science, University of Al-Qadisiyah, Iraq e-mail : abbas.kareem.w@qu.edu.iq
Abstract. In the present paper, we determine new characterisations of the sub-classes TS∗C(α, β; γ) and TCC(α, β; γ) of analytic functions associated with Pascal dis-tribution series Φmq (z) = z − P∞ n=2 n+m−2 m−1 q n−1
(1 − q)mzn. Further, we give neces-sary and sufficient conditions for an integral operator related to Pascal distribution series Gm
q f (z) = Rz
0 Φmq(t)
t dt to belong to the above classes. Several corollaries and consequences of the main results are also considered.
1. Introduction and Definitions
LetA denote the class of the normalized functions of the form
(1.1) f (z) = z +
∞ X
n=2 anzn,
which are analytic in the open unit disk U = {z ∈ C : |z| < 1} and are normalized by the conditions f (0) = f0(0)−1 = 0. Further, letT be the subclass of A consisting
* Corresponding Author.
Received May 20, 2020; revised October 7, 2020; accepted October 13, 2020. 2020 Mathematics Subject Classification: 30C45.
Key words and phrases: Analytic functions, Hadamard product, Pascal distribution series. ORCID of the first author is 0000-0001-8608-8063.
of functions of the form (1.2) f (z) = z − ∞ X n=2 |an| zn, z ∈ U.
A function f ∈A is said to be starlike of order α(0 ≤ α < 1) if it satisfies
(1.3) R zf
0(z) f (z)
> α (z ∈ U).
Also, we say that a function f ∈A is said to be convex of order α(0 ≤ α < 1) if it satisfies (1.4) R 1 + zf 00(z) f0(z) > α (z ∈ U).
We denote by S∗(α) andC(α) the classes of functions that starlike of order α and convex of order α in U, respectively. Further, TS∗(α) and TC(α) denote the sub-classes of T consisting of functions which are starlike of order α(0 ≤ α < 1) and convex of order α(0 ≤ α < 1) with negative coefficients in U, respectively [21].
Interesting generalization of the functions classes S∗(α) andC(α), are classes S∗(α, β) andC(α, β), where S∗(α, β) = f ∈A:R zf0(z) βf0(z) + (1 − β)f (z) > α, (α, β ∈ [0, 1), z ∈ U) and C(α, β) =f ∈A:R f 0(z) + zf00(z) f0(z) + βzf00(z) > α, (α, β ∈ [0, 1), z ∈ U) .
The classes TS∗(α, β) = S∗(α, β) ∩T and TC(α, β) = C(α, β) ∩ T were extensively studied by Altinta¸s and Owa [1], Porwal [19], Moustafa [13] and Porwal and Dixit [20].
Inspired by the studies mentioned above, Topkaya and Mustafa [23] defined a unification of the functions classesS∗(α, β) andC(α, β) as follows.
Definition 1.1. A function f of the form (1.1) is said to be in the classS∗C(α, β; γ) if it satisfies the following condition:
(1.5) R zf0(z) + γz2f00(z) γz (f0(z) + βzf00(z)) + (1 − γ) (βf0(z) + (1 − β)f (z)) > α (z ∈ U), where α, β ∈ [0, 1) and γ ∈ [0, 1]. Also we denote TS∗C(α, β; γ) = S∗C(α, β; γ) ∩ T.
Let f ∈ CC(α, β; γ) if and only if zf0 ∈ S∗C(α, β; γ) and denote the classTCC(α, β; γ) to be defined as
TCC(α, β; γ) = CC(α, β; γ) ∩ T.
In particular, the classTS∗C(α, β; 0) = TS∗(α, β) andTS∗C(α, β; 1) = TC(α, β). Also, we have TS∗C(α, 0; 0) = TS∗(α) andTS∗C(α, 0; 1) = TC(α).
A function f ∈A is said to be in the class Rτ
(A, B),τ ∈ C\{0}, −1 ≤ B < A ≤ 1, if it satisfies the inequality
f0(z) − 1 (A − B)τ − B[f0(z) − 1] < 1, z ∈ U. This class was introduced by Dixit and Pal [4].
A variable X is said to be Pascal distribution if it takes the values 0, 1, 2, 3, . . . with probabilities (1 − q)m,qm(1 − q) m 1! , q2m(m + 1)(1 − q)m 2! , q3m(m + 1)(m + 2)(1 − q)m 3! , . . . ,
respectively, where q and m are called the parameters, and thus P (X = r) =r + m − 1
m − 1
qr(1 − q)m, r = 0, 1, 2, 3, . . . .
Very recently, El-Deeb et al. [6] (see also, [2, 8, 9, 15]) introduced a power series whose coefficients are probabilities of Pascal distribution. Let
Ψmq (z) := z + ∞ X n=2 n + m − 2 m − 1 qn−1(1 − q)mzn, z ∈ U,
where m ≥ 1, 0 ≤ q ≤ 1. We note that, by the ratio test, the radius of convergence of above series is infinity. We also define the series
(1.6) Φmq (z) := 2z − Ψmq (z) = z − ∞ X n=2 n + m − 2 m − 1 qn−1(1 − q)mzn, z ∈ U.
Let consider the linear operatorIm
q :A → A defined by the Hadamard product Im q f (z) := Ψ m q (z) ∗ f (z) = z + ∞ X n=2 n + m − 2 m − 1 qn−1(1 − q)manzn, z ∈ U, where m ≥ 1 and 0 ≤ q ≤ 1.
There are several known results on connections between various subclasses of analytic and univalent functions using hypergeometric functions (see for example, [3, 11, 12, 22]) and using various distributions such as Yule-Simon distributions, Log-arithmic distributions, Poisson distributions, Binomial distributions, Beta-Binomial
distributions, Zeta distributions, Geometric distributions and the Bernoulli distri-bution (see, for example, [5, 7, 10, 14, 16, 17, 18]).
In this paper, we determine necessary and sufficient conditions for Φm q to be in the classes TS∗C(α, β; γ) and TCC(α, β; γ). Furthermore, we give sufficient conditions for Im
q (Rτ(A, B)) ⊂ TS
∗C(α, β; γ) and Im q (R
τ(A, B)) ⊂ TCC(α, β; γ). Finally, we provide necessary and sufficient conditions for the integral operator Gm
q f (z) = Rz
0 Φmq(t)
t dt to belong to the above classes. 2. Preliminary Lemmas
To establish our main results, we need the following lemmas.
Lemma 2.1.([23]) A function f ∈T of the form ( 1.2) is in the class TS∗C(α, β; γ) if and only if (2.1) ∞ X n=2 (1 + (n − 1)γ) (n − α − (n − 1)αβ) |an| ≤ 1 − α .
The result (2.1) is sharp.
Lemma 2.2. A function f ∈T of the form (1.2) is in the class TCC(α, β; γ) if and only if (2.2) ∞ X n=2 n(1 + (n − 1)γ) (n − α − (n − 1)αβ) |an| ≤ 1 − α .
The result (2.2) is sharp.
Lemma 2.3.([4]) If the function f ∈Rτ(A, B)is of the form (1.1), then |an| ≤ (A − B)
|τ |
n, n ∈ N − {1}. The result is sharp for the function
f (z) = Z z 0 (1 + (A − B) τ t n−1 1 + Btn−1)dt, (z ∈ U; n ∈ N − {1}).
3. Necessary and Sufficient Conditions
For convenience throughout in the sequel, we use the following identities that hold for m ≥ 1 and 0 ≤ q < 1:
∞ X n=0 n + m − 1 m − 1 qn= 1 (1 − q)m, ∞ X n=0 n + m − 2 m − 2 qn= 1 (1 − q)m−1, ∞ X n=0 n + m m qn= 1 (1 − q)m+1, ∞ X n=0 n + m + 1 m + 1 qn = 1 (1 − q)m+2.
By simple calculations we derive the following relations: ∞ X n=2 n + m − 2 m − 1 qn−1= ∞ X n=0 n + m − 1 m − 1 qn− 1 = 1 (1 − q)m− 1, (3.1) ∞ X n=2 (n − 1)n + m − 2 m − 1 qn−1= qm ∞ X n=0 n + m m qn= qm (1 − q)m+1, (3.2) ∞ X n=3 (n − 1)(n − 2)n + m − 2 m − 1 qn−1= q2m(m + 1) ∞ X n=0 n + m + 1 m + 1 qn = q 2m(m + 1) (1 − q)m+2. (3.3) and ∞ X n=4 (n − 1)(n − 2)(n − 3)n + m − 2 m − 1 qn−1 (3.4) = q3m(m + 1)(m + 2) ∞ X n=0 n + m + 1 m + 1 qn =q 3m(m + 1)(m + 2) (1 − q)m+3 .
Unless otherwise mentioned, we shall assume in this paper that α, β ∈ [0, 1), γ ∈ [0, 1] and 0 ≤ q < 1.
Firstly, we obtain the necessary and sufficient conditions for Φm
q to be in the classTS∗C(α, β; γ). Theorem 3.1. If m ≥ 1, then Φm q ∈TS ∗C(α, β; γ) if and only if (3.5) γ(1 − αβ)q 2m(m + 1) (1 − q)m+2 + (γ(2 − α) − αβ(γ + 1) + 1) qm (1 − q)m+1 ≤ 1 − α. Proof. Since Φmq is defined by (1.6), in view of Lemma 2.1 it is sufficient to show that P := ∞ X n=2 (1 + (n − 1)γ) (n − α − (n − 1)αβ)n + m − 2 m − 1 qn−1(1 − q)m ≤ 1 − α. Writing n = (n − 1) + 1 and n2= (n − 1)(n − 2) + 3(n − 1) + 1
and using (3.1)-(3.3), we get P = ∞ X n=2 γ(1 − αβ)n2+ (αβ(2γ − 1) − γ(1 + α) + 1) n + α(1 − γ)(β − 1) ×n + m − 2 m − 1 qn−1(1 − q)m = γ(1 − αβ) ∞ X n=3 (n − 1)(n − 2)n + m − 2 m − 1 qn−1(1 − q)m + (γ(2 − α) − αβ(γ + 1) + 1) ∞ X n=2 (n − 1)n + m − 2 m − 1 qn−1(1 − q)m + (1 − α) ∞ X n=2 n + m − 2 m − 1 qn−1(1 − q)m = γ(1 − αβ)q 2m(m + 1) (1 − q)2 + (γ(2 − α) − αβ(γ + 1) + 1) qm (1 − q) + (1 − α) (1 − (1 − q)m).
but this last expression is upper bounded by 1 − α if and only if (3.5) holds. 2 Theorem 3.2. If m ≥ 1,then Φmq ∈TCC(α, β; γ) if and only if
γ(1 − αβ)q 3m(m + 1)(m + 2) (1 − q)m+3 + (γ(5 − α) − αβ (4γ + 1) + 1) q2m(m + 1) (1 − q)m+2 + (2γ(2 − α) − 2αβ (γ + 1) + 3 − α) qm (1 − q)m+1 ≤ 1 − α. (3.6)
Proof. In view of Lemma 2.2 it is sufficient to show that
Q := ∞ X n=2 n(1 + (n − 1)γ) (n − α − (n − 1)αβ)n + m − 2 m − 1 qn−1(1 − q)m ≤ 1 − α. Writing n = (n − 1) + 1 n2= (n − 1)(n − 2) + 3(n − 1) + 1 and n3= (n − 1)(n − 2)(n − 3) + 6(n − 1)(n − 2) + 7(n − 1) + 1
and using (3.1)-(3.5), we get Q = ∞ X n=2 γ(1 − αβ)n3+ (αβ(2γ − 1) − γ(1 + α) + 1) n2+ α(1 − γ)(β − 1)n ×n + m − 2 m − 1 qn−1(1 − q)m = γ(1 − αβ) ∞ X n=4 (n − 1)(n − 2)(n − 3)n + m − 2 m − 1 qn−1(1 − q)m + (γ(5 − α) − αβ (4γ + 1) + 1) ∞ X n=3 (n − 1)(n − 2)n + m − 2 m − 1 qn−1(1 − q)m + (2γ(2 − α) − 2αβ (γ + 1) + 3 − α) ∞ X n=2 (n − 1)n + m − 2 m − 1 qn−1(1 − q)m + (1 − α) ∞ X n=2 n + m − 2 m − 1 qn−1(1 − q)m = γ(1 − αβ)q 3m(m + 1)(m + 2) (1 − q)3 + (γ(5 − α) − αβ (4γ + 1) + 1) q2m(m + 1) (1 − q)2 + (2γ(2 − α) − 2αβ (γ + 1) + 3 − α) qm (1 − q)+ (1 − α) (1 − (1 − q) m).
but this last expression is upper bounded by 1 − α if and only if (3.6) holds. 2 4. Inclusion Properties
Making use of Lemma 2.3, we will study the action of the Pascal distribution series on the classesTS∗C(α, β; γ) and TCC(α, β; γ).
Theorem 4.1. Let m > 1. If f ∈Rτ(A, B), thenImq f (z) is inTS
∗C(α, β; γ) if (A − B)|τ | γ(1 − αβ) qm (1 − q)+ (αβ(γ − 1) − γα + 1) (1 − (1 − q) m) +α(1 − γ)(β − 1) q(m − 1) [(1 − q) − (1 − q) m− q(m − 1)(1 − q)m] ≤ 1 − α. (4.1)
Proof. In view of Lemma 2.1, it suffices to show that
Λ := ∞ X n=2 (1 + (n − 1)γ) (n − α − (n − 1)αβ)n + m − 2 m − 1 qn−1(1 − q)m|an| ≤ 1 − α.
Since f ∈Rτ(A, B), then by Lemma 2.3, we have (4.2) |an| ≤ (A − B) |τ | n . Thus, we have Λ = (A − B)|τ | "∞ X n=2 [γ(1 − αβ)n + (αβ(2γ − 1) − γ(1 + α) + 1) +1 nα(1 − γ)(β − 1) n + m − 2 m − 1 qn−1(1 − q)m = (A − B)|τ | "∞ X n=2 [γ(1 − αβ)(n − 1) + (αβ(γ − 1) − γα + 1) +1 nα(1 − γ)(β − 1) n + m − 2 m − 1 qn−1(1 − q)m = (A − B)|τ | γ(1 − αβ) qm (1 − q)+ (αβ(γ − 1) − γα + 1) (1 − (1 − q) m) +α(1 − γ)(β − 1) q(m − 1) [(1 − q) − (1 − q) m− q(m − 1)(1 − q)m] .
But this last expression is bounded by 1 − α, if (4.1) holds. This completes the
proof of Theorem 4.1. 2
Theorem 4.2. Let m ≥ 1. If f ∈Rτ(A, B), then Imq f ∈TCC(α, β; γ) if (A − B)|τ | γ(1 − αβ)q 2m(m + 1) (1 − q)m+2 + (γ(2 − α) − αβ(γ + 1) + 1) qm (1 − q)m+1 ≤ 1 − α.
Proof. According to Lemma 2.2 it is sufficient to show that H := ∞ X n=2 n(1 + (n − 1)γ) (n − α − (n − 1)αβ)n + m − 2 m − 1 qn−1(1 − q)m|an| ≤ 1 − α.
Since f ∈Rτ(A, B), using Lemma we have |an| ≤ (A − B) |τ | n , n ∈ N \ {1}, therefore H ≤ (A−B) |τ | "∞ X n=2 (1 + (n − 1)γ) (n − α − (n − 1)αβ)n + m − 2 m − 1 qn−1(1 − q)m #
The remaining part of the proof is similar to that of Theorem 3.1 and so we omit
the details. 2
5. An Integral Operator
Theorem 5.1. Let m ≥ 1. Then the integral operator
(5.1) Gmq (z) := Z z 0 Φm q (t) t dt, z ∈ U, is in TCC(α, β; γ) if and only if the inequality (3.5) holds. Proof. According to (1.6) it follows that
Gm q (z) = z − ∞ X n=2 n + m − 2 m − 1 qn−1(1 − q)mz n n, z ∈ U. Using Lemma 2.2, the functionGm
q (z) belongs toTCC(α, β; γ) if and only if ∞ X n=2 n(1 + (n − 1)γ) (n − α − (n − 1)αβ)1 n n + m − 2 m − 1 qn−1(1 − q)m≤ 1 − α. By a similar proof like those of Theorem 3.1 we get thatGm
q ∈TCC(α, β; γ) if and
only if (3.5) holds. 2
Theorem 5.2. If m > 1, then the integral operator Gm
q (z) given by (5.1) is in TS∗C(α, β; γ) if and only if γ(1 − αβ) qm (1 − q)+ (αβ(γ − 1) − γα + 1) (1 − (1 − q) m) +α(1 − γ)(β − 1) q(m − 1) [(1 − q) − (1 − q) m− q(m − 1)(1 − q)m] ≤ 1 − α. (5.2)
The proof of Theorem Theorem 5.2 is lines similar to the proof of Theorem 4.1, so we omitted the proof of Theorem 5.2.
6. Corollaries and Consequences
By taking γ = 0 and β = 1 in Theorems 3.1, 4.1 and 5.2, we obtain the following necessary and sufficient conditions for Pascal distribution series to be in the classes TS∗(α, β) andTC(α, β). Corollary 6.1. If m ≥ 1, then Φm q ∈TS ∗(α, β) if and only if (1 − αβ) qm (1 − q)m+1 ≤ 1 − α.
Corollary 6.2. If m ≥ 1, then Φm q ∈TC(α, β), if and only if (1 − αβ)q 2m(m + 1) (1 − q)m+2 + (3 − α − 2αβ) qm (1 − q)m+1 ≤ 1 − α. Corollary 6.3. Let m > 1. If f ∈Rτ(A, B), thenImq f (z) is inTS∗(α, β) if
(A − B)|τ | [(1 − αβ) (1 − (1 − q)m) +α(β − 1) q(m − 1)[(1 − q) − (1 − q) m − q(m − 1)(1 − q)m] ≤ 1 − α.
Corollary 6.4. Let m > 1. If f ∈Rτ(A, B), thenIm
q f (z) is inTC(α, β) if (A − B)|τ | (1 − αβ) qm (1 − q)+ (1 − α) (1 − (1 − q) m) ≤ 1 − α.
Corollary 6.5. If m > 1, then the integral operator Gmq (z) given by (5.1) is in TS∗ (α, β) if and only if (1 − αβ) (1 − (1 − q)m) + α(β − 1) q(m − 1)[(1 − q) − (1 − q) m− q(m − 1)(1 − q)m] ≤ 1 − α.
Corollary 6.6. If m > 1, then the integral operator Gmq (z) given by (5.1) is in TC(α, β) if and only if
(1 − αβ) qm
(1 − q)+ (1 − α) (1 − (1 − q)
m) ≤ 1 − α.
Acknowledgements. The authors would like to thank the referees for their help-ful comments and suggestions.
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