Coefficient inequalities for certain subclasses of multivalent functions associated with conic domain

Lemma 1 ([22]) Let ψ be a function of the form ψ(z) = 1 + ∞  n=1 ψnzn subordinate to a function H of the form H(z) = 1 + ∞  n=1 Cnzn. In particular, when H is univalent in the unit disk U and H(U) is convex, then |ψn| ≤|C1| (n ∈N).

Lemma 2 Suppose that the sequence {ak}∞ k=0 is defined by ap = 1 and an+p = δk(q + 1)[n + 1]q! 2[n + p]q!([n + 1]q – 1) n–1  l=1 [p + l]q! [l + 1]q!ap+l. (2.1) Then an+p–1 =

Theorem 1 Let f be a p-valently analytic function of the form (1.1). Then f belongs to the class k-S∗ q if it satisfies the condition ∞  n=1 [n + p]q! [n + 1]q!Λ3|an+p| < (1 + q), (3.1) where Λ3 = 2(k + 1)q  [n + 1]q – 1  +

Corollary 1 (See [11]) Any function f ∈A of the form (1.1) belongs to the class k-S∗T if it satisfies the inequality ∞  n=1  (k + 1)n + 1  |an| < 1.

Theorem 2 A function f ∈A(p) of the form (1.1) belongs to the function class k-Cq(p) if ∞  n=2 [n + p]q! [n]q! Λ3|an+p| < (1 + q), where Λ3 is defined in (3.2).

Theorem 3 A function f ∈A(p) having series expansion (1.1) belongs to the class k-K∗ q(p) if ∞  n=2  2(k + 1)Λ1 + Λ2  < (1 + q), (3.4) where Λ1 = bn+p – [n + 1]qan+p 

Theorem 4 A function f from the class A(p) having series expansion (1.1) belongs to the class k-C∗ q(p) if ∞  n=1 [n + 1]q  2(k + 1)Λ1 + Λ2  < (1 + q), where Λ1 and Λ2 are respectively presented in (3.5) and (3.6).

Theorem 5 Let f ∈k-S∗ q(p) be of the form (1.1). Then |an+p–1| ≤ n  j=2 [j]q{2([j – 1]q – 1) + (q + 1)δk} 2{[j]q – 1}[j + p – 1]q  n ∈N \ {1}  . (3.8)

Corollary 2 (See [11]) For an analytic function f ∈k-S∗T , we have |an| ≤ n–2  j=0 j(j – 2) + δk (j – 1)(j)  n ∈N \ {1}  .

Theorem 6 Let f ∈k-Cq(p) be of the form (1.1). Then |an+p–1| ≤ 1 [n + p]q n  j=2 [j]q{2([j – 1]q – 1) + (q + 1)δk} 2{[j]q – 1}[j + p – 1]q  n ∈N \ {1}  .

Theorem 7 Let f ∈k-K∗ q be of the form (1.1). Then |an+p–1| ≤ [n]q! [n + p]q! n–2  j=0 [j]q{2([j – 1]q – 1) + δk(q + 1)} 2{[j]q – 1}[j + p – 1]q  + (q + 1)|δk| 2[n]q n–1 

Corollary 3 (See [18]) Let f ∈k-UK be of the form (1.1). Then |an| ≤(|δk|)n–1 n! + |δk| n n–1  j=1 (|δk|)j–1 (j – 1)!  n ∈N \ {1}  . Further, setting

Corollary 4 ([12]) Let f ∈K be an analytic function. Then |an| ≤n  n ∈N \ {1}  .

Theorem 8 Let f ∈k-C∗ q(p) with series expansion (1.1). Then |an| ≤ [n]2 q! [n + p]2q! n–2  j=0 [j]q{2([j – 1]q – 1) + δk(q + 1)} 2{[j]q – 1}[j + p – 1]q  + (q + 1)|δk| 2[n + p]q! ·