Abstract
In the present investigation we define a new class of meromorphic functions on the punctured unit disk U∗:= {z ∈
C : 0 < |z| < 1} by making use of the Srivastava-Attiya operator. Coefficient inequalities, growth and distortion
inequalities, as well as radii of meromorphically starlikeness are obtained. We also establish some results concerning
the convolution products and inclusion results.
Results & Lemmas (13)
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.
Theorem 2.1.
Theorem 2.1. Let f ∈Mp, then f is in the class Ms b(α, β) if and only if ∞ X n=1 dn|an| ≤p(1 −αβ)ap (13) where dn = [n −p(1 −αβ)]Cs b(n)…
Theorem 2.1. Let f ∈Mp, then f is in the class Ms b(α, β) if and only if ∞ X n=1 dn|an| ≤p(1 −αβ)ap (13) where dn = [n −p(1 −αβ)]Cs b(n) (14) and α > 1 2 + β ;
Theorem 2.2.
Theorem 2.2. (Coefficient Estimate) Let f ∈Ms b(α, β), then ∞ X n=1 |an| ≤p(1 −αβ)ap dn. (16)
Theorem 2.2. (Coefficient Estimate) Let f ∈Ms b(α, β), then ∞ X n=1 |an| ≤p(1 −αβ)ap dn . (16)
Theorem 2.3.
Theorem 2.3. Let f ∈Ms b(α, β, ρ), then ∞ X n=1 |an| ≤ p(1 −αβ) dn + p(1 −αβ)ρn. (17)
Theorem 2.3. Let f ∈Ms b(α, β, ρ), then ∞ X n=1 |an| ≤ p(1 −αβ) dn + p(1 −αβ)ρn . (17)
Theorem 2.4.
Theorem 2.4. (Distortion Bounds) If f ∈Ms b(α, β, ρ), then µd1 −p(1 −αβ)r d1 + p(1 −αβ)ρ ¶ r−p ≤|f(z)| ≤ µd1 + p(1 + αβ)r d1 + p(1 −αβ)ρ ¶…
Theorem 2.4. (Distortion Bounds) If f ∈Ms b(α, β, ρ), then µd1 −p(1 −αβ)r d1 + p(1 −αβ)ρ ¶ r−p ≤|f(z)| ≤ µd1 + p(1 + αβ)r d1 + p(1 −αβ)ρ ¶ r−p, (0 < |z| = r < 1).
Theorem 2.5.
Theorem 2.5. If f ∈Ms b(α, β), then µ 1 −p(1 −αβ)r d1 ¶ r−p ≤|f(z)| ≤ µ 1 + p(1 + αβ)r d1 ¶ r−p, (0 < |z| = r < 1).
Theorem 2.5. If f ∈Ms b(α, β), then µ 1 −p(1 −αβ)r d1 ¶ r−p ≤|f(z)| ≤ µ 1 + p(1 + αβ)r d1 ¶ r−p, (0 < |z| = r < 1).
Theorem 3.1.
Theorem 3.1. Let the function f(z) defined by (1) be in the class Ms b(α, β), then we have f(z) is meromorphically p-valent starlike of…
Theorem 3.1. Let the function f(z) defined by (1) be in the class Ms b(α, β), then we have f(z) is meromorphically p-valent starlike of order µ(0 ≤µ < p) in the disc |z| < r1, that is, ℜ µ −zf ′(z) f(z) ¶ > µ, |z| < r1; 0 ≤µ < p; p ∈N, where |z| ≤ µ dn(p −µ) p(n −p + µ)(1 −αβ)
Theorem 4.1.
Theorem 4.1. For the function fj(z) (j = 1, 2) defined by (22)be in the class Ms b(α, β).Then (f1 ∗f2)(z) ∈ Ms b(α, δ) where δ ≤1 α µ 1 −…
Theorem 4.1. For the function fj(z) (j = 1, 2) defined by (22)be in the class Ms b(α, β).Then (f1 ∗f2)(z) ∈ Ms b(α, δ) where δ ≤1 α µ 1 − p(1 −αβ)2Cs b(1) p2(1 −αβ)2Cs b(1) + d2 1 ¶
Theorem 4.2.
Theorem 4.2. For functions f1(z) ∈Ms b(α, β) and f2(z) ∈Ms b(α, γ), then (f1 ∗f2)(z) ∈Ms b(α, ζ) where ζ ≤ (1 + αβ)(1 + αγ)Cs b(1) −(1…
Theorem 4.2. For functions f1(z) ∈Ms b(α, β) and f2(z) ∈Ms b(α, γ), then (f1 ∗f2)(z) ∈Ms b(α, ζ) where ζ ≤ (1 + αβ)(1 + αγ)Cs b(1) −(1 −αβ)(1 −αγ) α[(1 −αβ)(1 −αγ) + (1 + αβ)(1 + αγ)Cs b(1)]
Theorem 4.3.
Theorem 4.3. Let the functions fj(z)(j = 1, 2) defined by fj(z) = ap,iz−p + ∞ X n=1 |an,j|zn−p (j = 1, 2) be in the class Ms b(α, β, ρ) then…
Theorem 4.3. Let the functions fj(z)(j = 1, 2) defined by fj(z) = ap,iz−p + ∞ X n=1 |an,j|zn−p (j = 1, 2) be in the class Ms b(α, β, ρ) then the function h(z) defined by h(z) = (ap,1 + ap,2)z−p + ∞ X n=1 (|an,1|2 + |an,2|2)zn−p (37)
Theorem 5.1.
Theorem 5.1. Let the function f(z) given by (1) be in Ms b(α, δ). Then the integral operator F(z) = c Z 1 0 uc+p−1f(uz)du, (0 < u ≤1, 0 < c…
Theorem 5.1. Let the function f(z) given by (1) be in Ms b(α, δ). Then the integral operator F(z) = c Z 1 0 uc+p−1f(uz)du, (0 < u ≤1, 0 < c < ∞), is in Mp(α, δ) where δ = 1 α µ(c + n)(n −p + pαβ) −c(n −p)(1 −αβ) (c + n)(n −p + pαβ) + cp(1 −αβ) ¶ .
Theorem 5.2.
Theorem 5.2. Let f(z) given by (1), be in Ms b(α, β). Then F(z) = 1 c [(c + p)f(z) + zf ′(z)] = ap zp + ∞ X n=1 c + n c anzn−p, c > 0, (46)…
Theorem 5.2. Let f(z) given by (1), be in Ms b(α, β). Then F(z) = 1 c [(c + p)f(z) + zf ′(z)] = ap zp + ∞ X n=1 c + n c anzn−p, c > 0, (46) is in Ms b(α, β) for |z| ≤r(α, β, δ) where
Theorem 5.3.
Theorem 5.3. (Arithmetic Mean) Let the functions fi(z)(i = 1, 2,... µ) defined by fi(z) = ap,i zp + ∞ X n=1 an,izn−p, (an,i ≥0, i = 1, 2...,…
Theorem 5.3. (Arithmetic Mean) Let the functions fi(z)(i = 1, 2, . . . µ) defined by fi(z) = ap,i zp + ∞ X n=1 an,izn−p, (an,i ≥0, i = 1, 2..., µ, n ≥1) be in the class Ms b(α, β, ρ). Then the arithmetic mean of fi(z)(i = 1, 2, . . . µ) defined by h(z) = 1 µ µ X n=1 fi(z)
Theorem 5.4.
Theorem 5.4. (Weighted Mean) Let the functions fi(z)(i = 1, 2) defined by fi(z) = ap,i zp + ∞ X n=1 an,izn−p, (an,i ≥0, i = 1, 2) be in the…
Theorem 5.4. (Weighted Mean) Let the functions fi(z)(i = 1, 2) defined by fi(z) = ap,i zp + ∞ X n=1 an,izn−p, (an,i ≥0, i = 1, 2) be in the class Ms b(α, β, ρ). Then the weighted mean of fi(z)(i = 1, 2) defined by Wc(z) = 1 2 [(1 −c)f1(z) + (1 + c)f2(z)] (49) is also in the class Ms b(α, β, ρ).
Function classes studied:
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