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Abstract
In this paper, we introduce a new class k-US(q,γ ,m,p), γ ∈C\{0}, of multivalent
functions using a newly defined q-analogue of a Salagean type differential operator.
We investigate the coefficient problem, Fekete–Szego inequality, and some other
properties related to subordination. Relevant connections of the results presented
here with those obtained in the earlier work are also pointed out.
MSC: Primary 30C45; secondary 30C50
Results & Lemmas (15)
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Lemma 2.1
Lemma 2.1 ([25]) Let h(z) = ∞ n=1 hnzn ≺F(z) = ∞ n=1 dnzn in E. If F(z) is convex univalent in E, then |hn| ≤|d1|, n ≥1.
Lemma 2.1 ([25]) Let h(z) = ∞ n=1 hnzn ≺F(z) = ∞ n=1 dnzn in E. If F(z) is convex univalent in E, then |hn| ≤|d1|, n ≥1.
Lemma 2.2
Lemma 2.2 ([31]) Let k ∈[0,∞) and let hk,γ be defined (1.6). If hk,γ (z) = 1 + Q1z + Q2z2 + ···, (2.1) Q1 = ⎧ ⎪⎪⎨ ⎪⎪⎩ 2γ A2 1–k2, 0 ≤k < 1,…
Lemma 2.2 ([31]) Let k ∈[0,∞) and let hk,γ be defined (1.6). If hk,γ (z) = 1 + Q1z + Q2z2 + ··· , (2.1) Q1 = ⎧ ⎪⎪⎨ ⎪⎪⎩ 2γ A2 1–k2 , 0 ≤k < 1, 8γ π2 , k = 1, π2γ 4(1+t)√tK2(t)(k2–1),
Lemma 2.3
Lemma 2.3 ([18]) Let h(z) = 1 + ∞ n=1 cnzn be analytic in E and satisfy Re h(z) > 0 for z in E. Then the following sharp estimate holds:…
Lemma 2.3 ([18]) Let h(z) = 1 + ∞ n=1 cnzn be analytic in E and satisfy Re{h(z)} > 0 for z in E. Then the following sharp estimate holds: c2 – μc2 1 ≤2max 1,|2μ – 1| , ∀μ ∈C.
Theorem 3.1
Theorem 3.1 Let f (z) ∈k – US(q,γ,m,p). Then Sm q,pf (z) ≺zexp z 0 p hk,γ (w(z)) – 1 ζ dξ, (3.1) where w(z) is analytic in E with w(0) =…
Theorem 3.1 Let f (z) ∈k – US(q,γ ,m,p). Then Sm q,pf (z) ≺zexp z 0 p{hk,γ (w(z))} – 1 ζ dξ, (3.1) where w(z) is analytic in E with w(0) = 0 and |w(z)| < 1. Moreover, for |z| = ρ, we have exp 1 0 p{hk,γ (–ρ)} – 1 ρ
Corollary 3.2
Corollary 3.2 Let f (z) ∈k – US(q,γ,m). Then Sm q f (z) ≺zexp z 0 hk,γ (w(ξ)) – 1 ζ dξ, where w(z) is analytic in E with w(0) = 0 and…
Corollary 3.2 Let f (z) ∈k – US(q,γ ,m). Then Sm q f (z) ≺zexp z 0 hk,γ (w(ξ)) – 1 ζ dξ, where w(z) is analytic in E with w(0) = 0 and |w(z)| < 1. Moreover, for |z| = ρ, we have exp 1 0 hk,γ (–ρ) – 1 ρ dρ
Theorem 3.3
Theorem 3.3 If f (z) ∈k – US(q,γ,m,p), then |ap+1| ≤ δ [p + 1]q – p ψp+1 (3.6) and |an+p–1| ≤ δ [n + p – 1]q – p ψn+p–1 n–2 j=1
Theorem 3.3 If f (z) ∈k – US(q,γ ,m,p), then |ap+1| ≤ δ {[p + 1]q – p}ψp+1 (3.6) and |an+p–1| ≤ δ {[n + p – 1]q – p}ψn+p–1 n–2 j=1
Corollary 3.4
Corollary 3.4 ([12]) If f (z) ∈k – US(q,γ,m), then |a2| ≤ δ [2]q – 1 [2]m q and |an| ≤ δ [n]q – 1 [n]m q n–2 j=1
Corollary 3.4 ([12]) If f (z) ∈k – US(q,γ ,m), then |a2| ≤ δ {[2]q – 1}[2]m q and |an| ≤ δ {[n]q – 1}[n]m q n–2 j=1
Theorem 3.5
Theorem 3.5 Let 0 ≤k < ∞be fixed and let f (z) ∈k – US(q,γ,m,p) with the form (1.1). Then, for a complex number μ, ap+2 – μa2 p+1 ≤ pQ1…
Theorem 3.5 Let 0 ≤k < ∞be fixed and let f (z) ∈k – US(q,γ ,m,p) with the form (1.1). Then, for a complex number μ, ap+2 – μa2 p+1 ≤ pQ1 2[2p + 1]m q {[p + 2]q – p} max 1,|2v – 1|
Corollary 3.6
Corollary 3.6 ([12]) Let 0 ≤k < ∞be fixed and let f (z) ∈k – US(q,γ,m) with the form (1.1). Then, for a complex number μ, a3 – μa2 2 ≤…
Corollary 3.6 ([12]) Let 0 ≤k < ∞be fixed and let f (z) ∈k – US(q,γ ,m) with the form (1.1). Then, for a complex number μ, a3 – μa2 2 ≤ Q1 2[3]m q {[3]q – 1} max 1,|2v – 1|
Theorem 3.7
Theorem 3.7 If a function f (z) ∈Ap has the form (1.1) and satisfies the condition ∞ n=p+1 [n]q – p (k + 1) + p|γ | |ψn||an| ≤|γ…
Theorem 3.7 If a function f (z) ∈Ap has the form (1.1) and satisfies the condition ∞ n=p+1 [n]q – p (k + 1) + p|γ | |ψn||an| ≤|γ ||p|, (3.16) then f (z) ∈k – US(q,γ ,m,p).
Corollary 3.8
Corollary 3.8 ([12]) If a function f (z) ∈A has the form (1.1) and satisfies the condition ∞ n=2 [n]q – 1 (k + 1) + |γ | [n]m q…
Corollary 3.8 ([12]) If a function f (z) ∈A has the form (1.1) and satisfies the condition ∞ n=2 [n]q – 1 (k + 1) + |γ | [n]m q |an| ≤|γ |, then f (z) ∈k – US(q,γ ,m). When q →1, p = 1, m = 0, γ = 1–α, with 0 ≤α < 1, we have the following known result, proved by Shams et al. in [28].
Corollary 3.9
Corollary 3.9 A function f ∈A of the form (1.1) is in the class SD(k,α) if it satisfies the condition ∞ n=2 n(k + 1) – (k + α) |an| ≤1…
Corollary 3.9 A function f ∈A of the form (1.1) is in the class SD(k,α) if it satisfies the condition ∞ n=2 n(k + 1) – (k + α) |an| ≤1 – α, where 0 ≤α < 1 and k ≥0. When q →1, p = 1, m = 0, γ = 1 – α, with 0 ≤α < 1 and k = 0, we have the following known result proved by Silverman in [30].
Corollary 3.10
Corollary 3.10 A function f ∈A of the form (1.1) is in the class SD(α) if it satisfies the condition ∞ n=2 n – α |an| ≤1 – α.
Corollary 3.10 A function f ∈A of the form (1.1) is in the class SD(α) if it satisfies the condition ∞ n=2 {n – α}|an| ≤1 – α.
Theorem 3.11
Theorem 3.11 Let f (z) ∈k – US(q,γ,m,p). Then f (E) contains an open disk of radius r = [p + 1]q – p ψp+1 (p + 1) [p + 1]q – p ψp+1 + δ,…
Theorem 3.11 Let f (z) ∈k – US(q,γ ,m,p). Then f (E) contains an open disk of radius r = {[p + 1]q – p}ψp+1 (p + 1){[p + 1]q – p}ψp+1 + δ , where δ = p|Q1| with Q1 given by (2.2).
Corollary 3.12
Corollary 3.12 ([12]) Let f (z) ∈k –US(q,γ,m). Then f (E) contains an open disk of radius r = [2]q – 1 [2]m q 2[2]m q [2]q – 1 + Q1, where…
Corollary 3.12 ([12]) Let f (z) ∈k –US(q,γ ,m). Then f (E) contains an open disk of radius r = {[2]q – 1}[2]m q 2[2]m q {[2]q – 1} + Q1 , where Q1 is given by (2.2).
Function classes studied:
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