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Abstract
We consider two new subclasses S∑m(τ,λ,α) and S∑m(τ,λ,β) of ∑m consisting of analytic and m-fold symmetric bi-
univalent functions in the open unit disk U. Furthermore, we establish bounds for the coefficients of functions in these subclasses and
several related classes are also considered. In addition to these, connections to earlier known results are presented.
Results & Lemmas (11)
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Lemma 1.
Lemma 1. [16]. If p ∈P, then |pn| ≤2, (n ∈N = 1,2,... ) and p2 −p12 2 ≤2 −|p1|2 2, (10) © 2019 BISKA Bilisim Technology
Lemma 1. [16]. If p ∈P, then |pn| ≤2, (n ∈N = {1,2,...}) and p2 −p12 2 ≤2 −|p1|2 2 , (10) © 2019 BISKA Bilisim Technology
Theorem 1.
Theorem 1. Let fgiven by (7) be in the class S∑m(τ,λ,α), 0 < α ≤1. Then, |am+1| ≤ 2α |τ| q 2m(m+ 2m2λ −m2λ 2)ατ −(α −1)m2 (1 + mλ)2, (13)…
Theorem 1. Let fgiven by (7) be in the class S∑m(τ,λ,α), 0 < α ≤1. Then, |am+1| ≤ 2α |τ| q 2m(m+ 2m2λ −m2λ 2)ατ −(α −1)m2 (1 + mλ)2 , (13) |a2m+1| ≤2(m+ 1)α2τ2 m2 (1 + mλ)2 + α |τ| m(1 + 2mλ). (14)
Theorem 2.
Theorem 2. Let given by (7) be in class textbfS∑m(λ,τ,β), 0 ≤β < 1. Then, |am+1| ≤ s 2|τ|(1 −β) m(m+ 2m2λ −m2λ 2), (31) |a2m+1| ≤2(m+ 1)τ2…
Theorem 2. Let given by (7) be in class textbfS∑m(λ,τ,β), 0 ≤β < 1. Then, |am+1| ≤ s 2|τ|(1 −β) m(m+ 2m2λ −m2λ 2), (31) |a2m+1| ≤2(m+ 1)τ2 (1 −β)2 m2 (1 + mλ)2 + |τ|(1 −β) m(1 + 2mλ). (32)
Lemma 2.
Lemma 2. [20]. Let the function φ (z) = 1 + h1z+ h2z2 +...; z ∈U such that φ ∈pn (β) then, |hk| ≤n(1 −β); k ≥1.
Lemma 2. [20]. Let the function φ (z) = 1 + h1z+ h2z2 + ... ; z ∈U such that φ ∈pn (β) then, |hk| ≤n(1 −β); k ≥1.
Theorem 3.
Theorem 3. I f f ∈S∑m(λ,τ,β), then |am+1| ≤min (s n|τ|(1 −β) m(m+ 2m2λ −m2λ 2), n|τ|(1 −β) m(1 + mλ) ), (48) |a2m+1| ≤ (m+ 1)n|τ|(1 −β)…
Theorem 3. I f f ∈S∑m(λ,τ,β), then |am+1| ≤min (s n|τ|(1 −β) m(m+ 2m2λ −m2λ 2), n|τ|(1 −β) m(1 + mλ) ) , (48) |a2m+1| ≤ (m+ 1)n|τ|(1 −β) 2m(m+ 2m2λ −m2λ 2). (49)
Corollary 1.
Corollary 1. [2]. Let f given by (7) be in the class S α ∑m (0 < α ≤1). Then, |am+1| ≤ 2α m√α + 1 and |a2m+1| ≤α m + 2(m+ 1)a2 m2. (59)
Corollary 1. [2]. Let f given by (7) be in the class S α ∑m (0 < α ≤1). Then , |am+1| ≤ 2α m√α + 1 and |a2m+1| ≤α m + 2(m+ 1)a2 m2 . (59)
Corollary 2.
Corollary 2. [2]. Let f given by (7) be in the class Sβ ∑m (0 ≤β < 1). Then, |am+1| ≤ p 2(1 −β) m and |a2m+1| ≤2(m+ 1)(1 −β)2 m2 + 1 −β m.…
Corollary 2. [2]. Let f given by (7) be in the class Sβ ∑m (0 ≤β < 1). Then, |am+1| ≤ p 2(1 −β) m and |a2m+1| ≤2(m+ 1)(1 −β)2 m2 + 1 −β m . (60) Classes S α ∑mand Sβ
Corollary 3.
Corollary 3. [14]. Let f given by (7) b in class S∗ ∑(α) (0 < α ≤1). Then, |a2| ≤ 2α √α + 1 and |a3| ≤4α2 + α. (65)
Corollary 3. [14]. Let f given by (7) b in class S∗ ∑(α) (0 < α ≤1). Then, |a2| ≤ 2α √α + 1 and |a3| ≤4α2 + α. (65)
Corollary 4.
Corollary 4. [14]. Let f given by (7) be in the class S∗ ∑(β) (0 ≤α < 1). Then, |a2| ≤ p 2(1 −β) and |a3| ≤4(1 −β)2 + (1 −β). (66) If we…
Corollary 4. [14]. Let f given by (7) be in the class S∗ ∑(β) (0 ≤α < 1). Then, |a2| ≤ p 2(1 −β) and |a3| ≤4(1 −β)2 + (1 −β). (66) If we set λ = 0 , λ = 1 and τ = 1 in Theorem 1, then the classes S∑m(τ,λ,β) reduce to the class Sβ ∑m. Thus, we obtain the following corollaries.
Corollary 5.
Corollary 5. [20]. I f 1 + 1 τ h zf ′(z) f(z) −1 i ∈pn (β) and 1 + 1 τ h wg′(w) g(w) −1 i ∈pn (β) then, |a2| ≤min np
Corollary 5. [20]. I f 1 + 1 τ h zf ′(z) f(z) −1 i ∈pn (β) and 1 + 1 τ h wg′(w) g(w) −1 i ∈pn (β) then, |a2| ≤min np
Corollary 6.
Corollary 6. [20]. I f 1 + 1 τ h zf ′′(z) f ′(z) i ∈pn (β) and 1 + 1 τ h wg′′(w) g′(w) i ∈pn (β) then, |a2| ≤min (r
Corollary 6. [20]. I f 1 + 1 τ h zf ′′(z) f ′(z) i ∈pn (β) and 1 + 1 τ h wg′′(w) g′(w) i ∈pn (β) then, |a2| ≤min (r
Function classes studied:
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