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Abstract
In the present article, our aim is to examine some useful problems including
the convolution problem, sufficiency criteria, coefficient estimates and Fekete-Szeg¨o type
inequalities for a new subfamily of analytic and multivalent functions associated with circular
domain. In addition, we also define and study a Bernardi integral operator in its q-extension
for multivalent functions.
Results & Lemmas (10)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 2.1.
Lemma 2.1. [23] Let h (z) = 1 + ∞ P n=1 dnzn ≺K (z) = 1 + ∞ P n=1 knzn in D. If K (z) is convex univalent in D, then |dn| ≦|k1|, for n ≧1.
Lemma 2.1. [23] Let h (z) = 1 + ∞ P n=1 dnzn ≺K (z) = 1 + ∞ P n=1 knzn in D. If K (z) is convex univalent in D, then |dn| ≦|k1| , for n ≧1.
Lemma 2.2.
Lemma 2.2. Let W contain all functions w that are analytic in D which satisfying w (0) = 0 & |w(z)| < 1. If the function w ∈W given by w(z)…
Lemma 2.2. Let W contain all functions w that are analytic in D which satisfying w (0) = 0 & |w(z)| < 1. If the function w ∈W given by w(z) = ∞ X k=1 wkzk (z ∈D) . then for λ ∈C, we have w2 −λw2 1 ≦max {1; |λ|} , (2.1) and w3 + 1 4w1w2 + 1
Theorem 3.1.
Theorem 3.1. Let f ∈Ap has the series form (1.1) and satisfing the inequality given by ∞ X n=1 ∧n+p ([n + p, q] (1 −B) −[p, q] (1 −A))…
Theorem 3.1. Let f ∈Ap has the series form (1.1) and satisfing the inequality given by ∞ X n=1 ∧n+p ([n + p, q] (1 −B) −[p, q] (1 −A)) |an+p| ≦[p, q] (A −B) . (3.1) Then f ∈S∗ p (q, µ, A, B) .
Corollary 3.2.
Corollary 3.2. Let f ∈A be given by (1.1) and satisfy the inequality ∞ X n=2 ([n, q] (1 −B) −1 + A) |an| ≦A −B. Then the function f ∈S∗ q…
Corollary 3.2. Let f ∈A be given by (1.1) and satisfy the inequality ∞ X n=2 ([n, q] (1 −B) −1 + A) |an| ≦A −B. Then the function f ∈S∗ q [A, B]. By choosing q →1−in the last corollary, we get the known result proved by Ahuja [3] and furthermore for A = 1 −α and B = −1, we obtain the result for the family S∗(ξ) which was proved by Silverman [27].
Theorem 3.3.
Theorem 3.3. Let f ∈S∗ p (q, µ, A, B) be of the form (1.1). Then |ap+1| ≦ψ1 (A −B) ∧1+p, (3.2) and for n ≧2, |an+p| ≦(A −B) ψn ∧n+p n−1 Y…
Theorem 3.3. Let f ∈S∗ p (q, µ, A, B) be of the form (1.1) . Then |ap+1| ≦ψ1 (A −B) ∧1+p , (3.2) and for n ≧2, |an+p| ≦(A −B) ψn ∧n+p n−1 Y t=1 1 + [p, q] (A −B)
Theorem 3.4.
Theorem 3.4. Let f ∈S∗ p (q, µ, A, B) and be given by (1.1). Then for λ ∈C ap+2 −λa2 p+1 ≦(A −B)ψ2 Λp+2 1; |υ|, where υ is given by υ = (B…
Theorem 3.4. Let f ∈S∗ p (q, µ, A, B) and be given by (1.1) . Then for λ ∈C ap+2 −λa2 p+1 ≦(A −B)ψ2 Λp+2 {1; |υ|} , where υ is given by υ = (B −(A −B)ψ1) + Λp+2ψ2 1 Λ2 p+1ψ2 (A −B)λ. (3.9)
Theorem 3.5.
Theorem 3.5. Let f ∈S∗ p (q, µ, A, B) and be given by (1.1). Then ap+3 − q + 2 q2 + q + 1 Λ1+pΛ2+p Λ3+p ap+2ap+1 + 1 [3, q] Λ3 1+p Λ3+p a3…
Theorem 3.5. Let f ∈S∗ p (q, µ, A, B) and be given by (1.1) . Then ap+3 − q + 2 q2 + q + 1 Λ1+pΛ2+p Λ3+p ap+2ap+1 + 1 [3, q] Λ3 1+p Λ3+p a3 p+1
Theorem 3.6.
Theorem 3.6. Let f ∈Ap be given by (1.1). Then the function f is in the class S∗ p (q, µ, A, B), if and only if eiθ (B −[p, q] A) z …
Theorem 3.6. Let f ∈Ap be given by (1.1) . Then the function f is in the class S∗ p (q, µ, A, B) , if and only if eiθ (B −[p, q] A) z Lµ+p−1 q f (z) ∗ (N + 1) zp −qLzp+1 (1 −z) (1 −qz) ̸= 0, (3.12) for all
Theorem 3.8.
Theorem 3.8. If f is of the form (1.1) belongs to the family S∗ p (q, µ, A, B) and Fη,p (z) = zp + ∞ X n=1 bn+pzn+p, (3.19) where Fη,p is…
Theorem 3.8. If f is of the form (1.1) belongs to the family S∗ p (q, µ, A, B) and Fη,p (z) = zp + ∞ X n=1 bn+pzn+p, (3.19) where Fη,p is the integral operator given by (3.17) , then |bp+1| ≦ [η + p, q] [η + p + 1, q] ψ1 (A −B) ∧1+p and for n ≧2
Theorem 3.9.
Theorem 3.9. Let f ∈S∗ p (q, µ, A, B) and be given by (1.1). Also if Fη,p is the integral operator defined by (3.17) and is of the form…
Theorem 3.9. Let f ∈S∗ p (q, µ, A, B) and be given by (1.1) . Also if Fη,p is the integral operator defined by (3.17) and is of the form (3.19) , then for σ ∈C bp+2 −σb2 p+1 ≦ [η + p, q] [η + p + 2, q] (A −B)ψ2 Λp+2 {1; |υ|} , where υ = (B −(A −B)ψ1) + Λp+2ψ2 1 Λ2
Function classes studied:
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