Abstract
In the present paper, we introduce and study a subclass of analytic and univalent
functions associated with Beta negative binomial distribution series which is defined in
the open unit disk . We discuss some important geometric properties of this subclass,
like, coefficient estimates, extreme points and integral representation. Also, we obtain
results about integral mean associated with fractional integral.
Results & Lemmas (6)
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Lemma 1.1
Lemma 1.1 [4]. If and H are analytic in with ≺H, then for 1 > 0 and = 7![ 0 < 7 < 1 T ||] ^ W V_ ≤ T |H|] ^ W V_.
Lemma 1.1 [4]. If and H are analytic in with ≺H, then for 1 > 0 and = 7![\ 0 < 7 < 1 T ||] ^ W V_ ≤ T |H|] ^ W V_ .
Theorem 2.1.
Theorem 2.1. A function E-,0 / ∈ is in the class `b, c, 4, 5, 3 if and only if ?? −cb + 14/3 D25 D2 4 + 5/3 + 4 + 5 D2?…
Theorem 2.1. A function E-,0 / ∈ is in the class `b, c, 4, 5, 3 if and only if ?? −cb + 14/3 D25 D2 4 + 5/3 + 4 + 5 D2? −1!
Corollary 2.1.
Corollary 2.1. If E-,0 / ∈`b, c, 4, 5, 3, then
Corollary 2.1. If E-,0 / ∈`b, c, 4, 5, 3, then
Theorem 3.1.
Theorem 3.1. Let E-,0 / 2 = and E-,0 / = −4 + 5/3 + 4 + 5 D2? −1! b + 11 −c ?? −cb + 14/3 D25 D2 , ? ≥2.…
Theorem 3.1. Let E-,0 / 2 = and E-,0 / = −4 + 5/3 + 4 + 5 D2? −1! b + 11 −c ?? −cb + 14/3 D25 D2 , ? ≥2. Then E-,0 / ∈`b, c, 4, 5, 3 if and only if it can be expressed in the form E-,0 / = O E-,0 /
Theorem 4.1.
Theorem 4.1. Let E-,0 / ∈`b, c, 4, 5, 3. Then E-,0 / = T T !) |T b −c U2 + −2 U2K1 −a U2L VU2 Q W ~ VUVU• Q W Q W,
Theorem 4.1. Let E-,0 / ∈`b, c, 4, 5, 3. Then E-,0 / = T T !){ |T b −c}U2 + { −2 U2K1 −a}U2L VU2 Q W ~ VUVU• Q W Q W ,
Theorem 5.1.
Theorem 5.1. If E-,0 / ∈`b, c, 4, 5, 3 and suppose that E-,0 / is defined by E-,0 / = −4 + 5/3 + 4 + 5 D2? −1! b + 11 −c…
Theorem 5.1. If E-,0 / ∈`b, c, 4, 5, 3 and suppose that E-,0 / is defined by E-,0 / = −4 + 5/3 + 4 + 5 D2? −1! b + 11 −c ?? −cb + 14/3 D25 D2 , ? ≥2. 5.1 Also let † −‡ˆ.2 ‰
Definitions (2)
Def 1.1
Definition 1.1 [2]. The fractional integral of order O (O > 0) is defined for a function by PQDR = 1 SO T U −U2DR VU Q W,…
Definition 1.1 [2]. The fractional integral of order O (O > 0) is defined for a function by PQDR = 1 SO T U −U2DR VU Q W , 1.3
Def 1.2.
Definition 1.2. A function E-,0 / ∈ is said to be in the class `a, b, c, 4, 5, 3 if it satisfies d eE-,0 / f ## b eE-,0 / f
Definition 1.2. A function E-,0 / ∈ is said to be in the class `a, b, c, 4, 5, 3 if it satisfies d eE-,0 / f ## b eE-,0 / f
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