Certain subclasses of Spirallike univalent functions related with Pascal distribution series
aa r X i v : . [ m a t h . C V ] J u l CERTAIN SUBCLASSES OF SPIRALLIKE UNIVALENT FUNCTIONSRELATED WITH PASCAL DISTRIBUTION SERIES
G. MURUGUSUNDARAMOORTHY
Abstract.
The purpose of the present paper is to find the necessary and sufficientconditions for the subclasses of analytic functions associated with Pascal distribution to bein subclasses of spiral-like univalent functions and inclusion relations for such subclassesin the open unit disk D . Further, we consider the properties of integral operator relatedto Pascal distribution series. Several corollaries and consequences of the main results arealso considered.
Mathematics Subject Classification (2010): 30C45.
Keywords : Univalent, Spiral-like, Hadamard product, Pascal distribution series. Introduction and definitions
Denote by A the class of functions whose members are of the form f ( z ) = z + ∞ X n =2 a n z n , (1)which are analytic in the open unit disk D = { z ∈ C : | z | < } and normalized by theconditions f (0) = 0 = f ′ (0) − . Let S be subclass of A whose members are given by(1) and are univalent in D . For functions f ∈ S be given by (1) and g ∈ S given by g ( z ) = z + P ∞ n =2 b n z n , we define the Hadamard product (or convolution) of f and g by( f ∗ g )( z ) = z + ∞ X n =2 a n b n z n , z ∈ D . The two well known subclass of S , are namely the class of starlike and convex functions (fordetails see Robertson [17]). A function f ∈ S is said to be starlike of order γ (0 ≤ γ < , if and only if Re (cid:18) zf ′ ( z ) f ( z ) (cid:19) > γ ( z ∈ D ) . This function class is denoted by S ∗ ( γ ) . We also write S ∗ (0) =: S ∗ , where S ∗ denotes theclass of functions f ∈ A that f ( D ) is starlike with respect to the origin.A function f ∈ S is said to be convex of order γ (0 ≤ γ <
1) if and only ifRe (cid:18) zf ′′ ( z ) f ′ ( z ) (cid:19) > γ ( z ∈ D ) . This class is denoted by K ( γ ) . Further, K = K (0), the well-known standard class of convexfunctions. By Alexander’s relation, it is a known fact that f ∈ K ⇔ zf ′ ( z ) ∈ S ∗ . A function f ∈ S is said to be spirallike if ℜ (cid:18) e − iα zf ′ ( z ) f ( z ) (cid:19) > α with | α | < π and for all z ∈ D this class of spiral-like function was introducedby[20]. Also f ( z ) is convex spiral-like if zf ′ ( z ) is spiral-like. Due to Murugusundramoorthy[8,9], we consider subclasses of spiral-like functions as below: Definition 1.1.
For 0 ≤ ρ < , ≤ γ < S ( ξ, γ, ρ ) := (cid:26) f ∈ S : Re (cid:18) e iξ zf ′ ( z )(1 − ρ ) f ( z ) + ρzf ′ ( z ) (cid:19) > γ cos ξ, | ξ | < π , z ∈ D (cid:27) . By virtue of Alexander’s relation, we define the following subclass:
Definition 1.2.
For 0 ≤ ρ < , ≤ γ < K ( ξ, γ, ρ ) := (cid:26) f ∈ S : Re (cid:18) e iξ zf ′′ ( z ) + f ′ ( z ) f ′ ( z ) + ρzf ′′ ( z ) (cid:19) > γ cos ξ, | ξ | < π , z ∈ D (cid:27) . By specialising the parameter ρ = 0 we remark the following : Definition 1.3.
For 0 ≤ γ < S ( ξ, γ ) := (cid:26) f ∈ S : Re (cid:18) e iξ zf ′ ( z ) f ( z ) (cid:19) > γ cos ξ, | ξ | < π , z ∈ D (cid:27) and K ( ξ, γ ) := (cid:26) f ∈ S : Re (cid:18) e iξ (cid:20) zf ′′ ( z ) f ′ ( z ) (cid:21)(cid:19) > γ cos ξ, | ξ | < π , z ∈ D (cid:27) . Now we state the necessary sufficient conditions for f in the above classes. Lemma 1.4. [8, 9]
A function f ( z ) of the form (1) is in S ( ξ, γ, ρ ) if ∞ X n =2 [(1 − ρ )( n −
1) sec ξ + (1 − γ )(1 + nρ − ρ )] | a n |≤ − γ, (2) where | ξ | < π , ≤ ρ < , ≤ γ < . Lemma 1.5.
A function f ( z ) of the form (1) is in S ( ξ, γ, ρ ) if ∞ X n =2 n [(1 − ρ )( n −
1) sec ξ + (1 − γ )(1 + nρ − ρ )] | a n |≤ − γ, (3) where | ξ | < π , ≤ ρ < , ≤ γ < . Proof.
Using Alexander type Theorem which states that, If f ∈ K ( ξ, γ, ρ ) if and only if zf ′ ∈ S ( ξ, γ, ρ ) , Thus z + ∞ P n =2 na n z n is in K ( ξ, γ, ρ ) . Hence by wringing a n in Lemma 1.4by na n we get the desired result. (cid:3) Lemma 1.6.
A function f ( z ) of the form (1) is in S ( ξ, γ ) if ∞ X n =2 [( n −
1) sec ξ + (1 − γ )] | a n |≤ − γ, (4) where | ξ | < π , ≤ γ < . PIRALLIKE UNIVALENT FUNCTIONS ... 3
Lemma 1.7.
A function f ( z ) of the form (1) is in K ( ξ, γ ) if ∞ X n =2 n [( n −
1) sec ξ + (1 − γ )] | a n |≤ − γ, (5) where | ξ | < π , ≤ γ < . Definition 1.8.
A function f ∈ S is said to be in the class R τ ( ϑ, δ ), ( τ ∈ C \{ } , <ϑ ≤ δ < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 − ϑ ) f ( z ) z + ϑf ′ ( z ) − τ (1 − δ ) + (1 − ϑ ) f ( z ) z + ϑf ′ ( z ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < z ∈ U ) . The class R τ ( ϑ, δ ) was introduced earlier by Swaminathan [21](for special cases see thereferences cited there in). Lemma 1.9. [21] If f ∈ R τ ( ϑ, δ ) is of form (1) , then | a n | ≤ | τ | (1 − δ )1 + ϑ ( n − , n ∈ N \ { } . (6) The bounds given in (6) is sharp.
A variable x is said to be Pascal distribution if it takes the values 0 , , , , . . . withprobabilities(1 − q ) m , qm (1 − q ) m
1! , q m ( m + 1)(1 − q ) m
2! , q m ( m + 1)( m + 2)(1 − q ) m
3! , . . . respectively,where q and m are called the parameters, and thus P ( x = k ) = (cid:18) k + m − m − (cid:19) q k (1 − q ) m , k = 0 , , , , . . . . Lately, El-Deeb [4](also see [1]) introduced a power series whose coefficients are probabili-ties of Pascal distributionΘ mq ( z ) = z + ∞ X n =2 (cid:18) n + m − m − (cid:19) q n − (1 − q ) m z n , z ∈ D where m ≥
1; 0 ≤ q ≤ mq ( z ) : A → A defined by the convolution or Hadamard productΛ mq f ( z ) = Θ mq ( z ) ∗ f ( z ) = z + ∞ X n =2 (cid:18) n + m − m − (cid:19) q n − (1 − q ) m a n z n , z ∈ D . Inspired by earlier results on relations between different subclasses of analytic and uni-valent functions by using hypergeometric functions (see for example,[2, 6, 7, 18, 19, 21])and by the recent investigations related with distribution series (see for example, [1, 3, 4,5, 11, 10, 15, 14, 16], we obtain necessary and sufficient condition for the function Φ mq to bein the classes S ( ξ, γ, ρ ) and K ( ξ, γ, ρ ), and information regarding the images of functionsbelonging in R τ ( ϑ, δ ) by applying convolution operator. Finally, we provide conditions forthe integral operator G mq ( z ) = R z mq ( t ) t dt belonging to the above classes. G. MURUGUSUNDARAMOORTHY The necessary and sufficient conditions
In order to prove our main results, we will use the following notations, for m ≥ ≤ q < ∞ X n =0 (cid:18) n + m − m − (cid:19) q n = 1(1 − q ) m ; ∞ X n =0 (cid:18) n + mm (cid:19) q n = 1(1 − q ) m +1 and ∞ X n =0 (cid:18) n + m + 1 m + 1 (cid:19) q n = 1(1 − q ) m +2 . (7)By simple computation we get the following relations: ∞ X n =2 (cid:18) n + m − m − (cid:19) q n − = ∞ X n =0 (cid:18) n + m − m − (cid:19) q n − − q ) m − ∞ X n =2 ( n − (cid:18) n + m − m − (cid:19) q n − = qm ∞ X n =0 (cid:18) n + mm (cid:19) q n = qm (1 − q ) m +1 (9)and ∞ X n =2 ( n − n − (cid:18) n + m − m − (cid:19) q n − = q m ( m + 1) ∞ X n =0 (cid:18) n + m + 1 m + 1 (cid:19) q n = q m ( m + 1)(1 − q ) m +2 . (10) Theorem 2.1. If m > , then Θ mq ( z ) ∈ S ( ξ, γ, ρ ) if [(1 − ρ ) sec ξ + ρ (1 − γ )] qm (1 − q ) m +1 ≤ − γ. (11) Proof.
Since Θ mq ( z ) = z + ∞ X n =2 (cid:18) n + m − m − (cid:19) q n − (1 − q ) m z n . Using the Lemma 1.4, it suffices to show that ∞ X n =2 [(1 − ρ )( n −
1) sec ξ + (1 − γ )(1 + nρ − ρ )] ≤ − γ. (12) PIRALLIKE UNIVALENT FUNCTIONS ... 5
From (12) we let M ( ξ, γ, ρ ) = ∞ X n =2 [(1 − ρ ) sec ξ ( n −
1) + (1 − γ )(1 + nρ − ρ )] (cid:18) n + m − m − (cid:19) q n − (1 − q ) m = [(1 − ρ ) sec ξ + ρ (1 − γ )](1 − q ) m ∞ X n =2 ( n − × (cid:18) n + m − m − (cid:19) q n − + (1 − γ )(1 − q ) m ∞ X n =2 (cid:18) n + m − m − (cid:19) q n − = [(1 − ρ ) sec ξ + ρ (1 − γ )](1 − q ) m qm ∞ X n =0 (cid:18) n + mm (cid:19) q n + (1 − γ )(1 − q ) m ∞ X n =0 (cid:18) n + m − m − (cid:19) q n − ! = [(1 − ρ ) sec ξ + ρ (1 − γ )](1 − q ) m qm (1 − q ) m +1 + (1 − γ )(1 − q ) m (cid:18) − q ) m − (cid:19) = [(1 − ρ ) sec ξ + ρ (1 − γ )] qm (1 − q ) + (1 − γ ) (1 − (1 − q ) m ) . But M ( ξ, γ, ρ ) is bounded above by 1 − γ if and only if (11) holds. (cid:3) Theorem 2.2. If m > , then Θ mq ( z ) ∈ K ( ξ, γ, ρ ) if [(1 − ρ ) sec ξ + (1 − γ )] m ( m + 1) q (1 − q ) + [2(1 − ρ ) secξ + (1 − γ )(4 − ρ )] mq − q + [(1 − γ )(2 − ρ )] (1 − (1 − q ) m ) ≤ − γ. (13) Proof.
In view of Lemma 1.5, we have to show that ∞ X n =2 n [(1 − ρ )( n −
1) sec ξ + (1 − γ )(1 + nρ − ρ )] (cid:18) n + m − m − (cid:19) q n − (1 − q ) m ≤ − γ. (14) G. MURUGUSUNDARAMOORTHY
Writing n = ( n −
1) + 1 and n = ( n − n −
2) + 3( n −
1) + 1 . From (14), consider the expression M ( ξ, γ, ρ ) = ∞ X n =2 n [(1 − ρ )( n −
1) sec ξ + (1 − γ )(1 + nρ − ρ )] × (cid:18) n + m − m − (cid:19) q n − (1 − q ) m = [(1 − ρ ) sec ξ + (1 − γ )](1 − q ) m ∞ X n =2 n (cid:18) n + m − m − (cid:19) q n − − (1 − ρ )[ secξ − (1 − γ )(1 − q ) m ∞ X n =2 n (cid:18) n + m − m − (cid:19) q n − = [(1 − ρ ) sec ξ + (1 − γ )](1 − q ) m ∞ X n =2 ( n − n − (cid:18) n + m − m − (cid:19) q n − + [2(1 − ρ ) secξ + (1 − γ )(4 − ρ )] (1 − q ) m ∞ X n =2 ( n − (cid:18) n + m − m − (cid:19) q n − + [(1 − γ )(2 − ρ )] (1 − q ) m ∞ X n =2 (cid:18) n + m − m − (cid:19) q n − = [(1 − ρ ) sec ξ + (1 − γ )](1 − q ) m q m ( m + 1) ∞ X n =0 (cid:18) n + m + 1 m + 1 (cid:19) q n + [2(1 − ρ ) secξ + (1 − γ )(4 − ρ )] (1 − q ) m mq ∞ X n =0 (cid:18) n + mm (cid:19) q n + [(1 − γ )(2 − ρ )] (1 − q ) m " ∞ X n =0 (cid:18) n + m − m − (cid:19) q n − . Now by using (8)-(10), we get M ( ξ, γ, ρ ) = [(1 − ρ ) sec ξ + (1 − γ )] m ( m + 1) q (1 − q ) + [2(1 − ρ ) secξ + (1 − γ )(4 − ρ )] mq − q + [(1 − γ )(2 − ρ )] (1 − (1 − q ) m ) . Hence, M ( ξ, γ, ρ ) is bounded above by 1 − γ if (13) is satisfied. (cid:3) Inclusion Properties
Making use of the Lemma 1.9, we will focus the influence of the Pascal distributionseries on the classes S ( ξ, γ, ρ ) and K ( ξ, γ, ρ ). PIRALLIKE UNIVALENT FUNCTIONS ... 7
Theorem 3.1. If f ∈ R τ ( ϑ, δ ) then Λ mq f ( z ) is in S ( ξ, γ, ρ ) if and only if | τ | (1 − δ ) ϑ n [(1 − ρ ) secξ + ρ (1 − γ )] [1 − (1 − q ) m ]+ (1 − ρ )(1 − γ − secξ ) q ( m −
1) [(1 − q ) − (1 − q ) m − q ( m − − q ) m ] (cid:27) ≤ − γ. (15) Proof.
In view of Lemma 1.4, it is required to show that ∞ X n =2 [(1 − ρ )( n −
1) sec ξ + (1 − γ )(1 + nρ − ρ )] (cid:18) n + m − m − (cid:19) q n − (1 − q ) m | a n |≤ − γ. Let M ( ξ, γ, ρ ) = ∞ X n =2 [(1 − ρ )( n −
1) sec ξ + (1 − γ )(1 + nρ − ρ )] × (cid:18) n + m − m − (cid:19) q n − (1 − q ) m | a n | . Since f ∈ R τ ( ϑ, δ ) , then by Lemma 1.9, we have | a n | ≤ | τ | (1 − δ )1 + ϑ ( n − , n ∈ N \ { } and 1 + ϑ ( n − ≥ ϑn. Thus,we have M ( ξ, γ, ρ ) ≤ | τ | (1 − δ ) ϑ " ∞ X n =2 n [(1 − ρ )( n −
1) sec ξ + (1 − γ )(1 + nρ − ρ )] × (cid:18) n + m − m − (cid:19) q n − (1 − q ) m (cid:21) = 2 | τ | (1 − δ ) ϑ (1 − q ) m " ∞ X n =2 [(1 − ρ ) secξ + ρ (1 − γ )]+ (1 − ρ )(1 − γ − secξ ) 1 n (cid:21) (cid:18) n + m − m − (cid:19) q n − (cid:21) . Using (8), we get M ( ξ, γ, ρ ) = 2 | τ | (1 − δ ) ϑ (1 − q ) m ( [(1 − ρ ) secξ + ρ (1 − γ )] " ∞ X n =0 (cid:18) n + m − m − (cid:19) q n − + (1 − ρ )(1 − γ ) q ( m − " ∞ X n =0 (cid:18) n + m − m − (cid:19) q n − − ( m − q = 2 | τ | (1 − δ ) ϑ n [(1 − ρ ) secξ + ρ (1 − γ )] [1 − (1 − q ) m ]+ (1 − ρ )(1 − γ − secξ ) q ( m −
1) [(1 − q ) − (1 − q ) m − q ( m − − q ) m ] (cid:27) . But M ( ξ, γ, ρ ) is bounded by 1 − γ , if (15) holds. This completes the proof of Theorem3.1. (cid:3) G. MURUGUSUNDARAMOORTHY
Applying Lemma 1.5 and using the same technique as in the proof of Theorem 2.2, wehave the following result.
Theorem 3.2. If f ∈ R τ ( ϑ, δ ) , then Λ mq f ( z ) is in K ( ξ, γ, ρ ) if and only if | τ | (1 − δ ) ϑ (cid:20) [(1 − ρ ) sec ξ + (1 − γ )] m ( m + 1) q (1 − q ) + [2(1 − ρ ) secξ + (1 − γ )(4 − ρ )] mq − q + [(1 − γ )(2 − ρ )] (1 − (1 − q ) m ) (cid:21) ≤ − γ. An integral operator
Theorem 4.1.
If the function G mq ( z ) is given by G mq ( z ) = Z z Θ mq ( t ) t dt, z ∈ D (16) then G mq ( z ) ∈ K ( ξ, γ, ρ ) if and only if [(1 − ρ ) sec ξ + ρ (1 − γ )] qm (1 − q ) m +1 ≤ − γ. Proof.
Since G mq ( z ) = z + ∞ X n =2 (cid:18) n + m − m − (cid:19) q n − (1 − q ) m z n n then by Lemma 1.5, we need only to verify that ∞ X n =2 n [(1 − ρ )( n −
1) sec ξ + (1 − γ )(1 + nρ − ρ )] × n (cid:18) n + m − m − (cid:19) q n − (1 − q ) m ≤ − γ, or, equivalently ∞ X n =2 [(1 − ρ )( n −
1) sec ξ + (1 − γ )(1 + nρ − ρ )] (cid:18) n + m − m − (cid:19) q n − (1 − q ) m ≤ − γ. The remaining part of the proof of Theorem 4.1 is similar to that of Theorem 2.1, and sowe omit the details. (cid:3)
Theorem 4.2. If m > , then the integral operator G mq given by (16) is in S ( ξ, γ, ρ ) if andonly if [(1 − ρ ) secξ + ρ (1 − γ )] [1 − (1 − q ) m ]+ (1 − ρ )(1 − γ − secξ ) q ( m −
1) [(1 − q ) − (1 − q ) m − q ( m − − q ) m ] ≤ − γ. Proof.
Since G mq ( z ) = z + ∞ X n =2 (cid:18) n + m − m − (cid:19) q n − (1 − q ) m z n n then by Lemma 1.4, we need only to verify that ∞ X n =2 n [(1 − ρ )( n −
1) sec ξ + (1 − γ )(1 + nρ − ρ )] (cid:18) n + m − m − (cid:19) q n − (1 − q ) m ≤ − γ. PIRALLIKE UNIVALENT FUNCTIONS ... 9
Thus, we have M ( ξ, γ, ρ ) = ∞ X n =2 n [(1 − ρ )( n −
1) sec ξ + (1 − γ )(1 + nρ − ρ )] × . (cid:18) n + m − m − (cid:19) q n − (1 − q ) m = (1 − q ) m " ∞ X n =2 [(1 − ρ ) secξ + ρ (1 − γ )]+ (1 − ρ )(1 − γ − secξ ) 1 n (cid:21) (cid:18) n + m − m − (cid:19) q n − (cid:21) . Using (8), we get M ( ξ, γ, ρ ) = (1 − q ) m ( [(1 − ρ ) secξ + ρ (1 − γ )] " ∞ X n =0 (cid:18) n + m − m − (cid:19) q n − + (1 − ρ )(1 − γ − secξ ) q ( m − " ∞ X n =0 (cid:18) n + m − m − (cid:19) q n − − ( m − q = n [(1 − ρ ) secξ + ρ (1 − γ )] [1 − (1 − q ) m ]+ (1 − ρ )(1 − γ − secξ ) q ( m −
1) [(1 − q ) − (1 − q ) m − q ( m − − q ) m ] (cid:27) . But M ( ξ, γ, ρ ) is bounded by 1 − γ , if (15) holds. This completes the proof of Theorem3.1. (cid:3) The proof of Theorem 4.2 is similar to that of Theorem 4.1, so we omitted the proof ofTheorem 4.2. 5.
Corollaries and consequences
By taking ρ = 0 in Theorems 2.1-4.2, we obtain the necessary and sufficient condition forPascal distribution series be in the classes S ( ξ, γ ) and K ( ξ, γ ) from the following corollaries. Corollary 5.1. If m > , then Θ mq is in S ( ξ, γ ) if and only if qm sec ξ (1 − q ) m +1 ≤ − γ. Corollary 5.2. If m > , then Θ mq is in K ( ξ, γ ) if and only if [sec ξ + (1 − γ )] m ( m + 1) q (1 − q ) + [2 secξ + 4(1 − γ )] mq − q + [2(1 − γ )] (1 − (1 − q ) m ) ≤ − γ. Corollary 5.3. If f ∈ R τ ( ϑ, δ ) then Λ mq is in S ( ξ, γ ) if and only if | τ | (1 − δ ) ϑ n secξ [1 − (1 − q ) m ]+ (1 − γ − secξ ) q ( m −
1) [(1 − q ) − (1 − q ) m − q ( m − − q ) m ] (cid:27) ≤ − γ. Corollary 5.4. If f ∈ R τ ( ϑ, δ ) , then Λ mq is in K ( ξ, γ ) if and only if | τ | (1 − δ ) ϑ (cid:20) [sec ξ + (1 − γ )] m ( m + 1) q (1 − q ) + [2 secξ + 4(1 − γ )] mq − q + [2(1 − γ )] (1 − (1 − q ) m ) i ≤ − γ. Corollary 5.5. If m > , then the integral operator G mq ( z ) given by (16) is in K ( ξ, γ ) ifand only if qm sec ξ (1 − q ) m +1 ≤ − γ. Corollary 5.6. If m > , then the integral operator G mq ( z ) given by (16) is in S ( ξ, γ ) ifand only if secξ [1 − (1 − q ) m ]+ (1 − γ − secξ ) q ( m −
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