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Abstract
The object of this paper is studying some properties of meromorphic functions which satisfy in the condition \[Re(zf(z)) > α|z^2f'(z)+zf(z)| .\] Parallel results for some related classes are also obtained.
Results & Lemmas (10)
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Theorem 2.1
Theorem 2.1 ME(α) ⊂MF(1 −1 α) ⊂ ∗ X (1 −1 α) 1 ≤α. If α = 1 all inclusions are proper and for α > 1 the result is sharp.
Theorem 2.1 ME(α) ⊂MF(1 −1 α) ⊂ ∗ X (1 −1 α) 1 ≤α. If α = 1 all inclusions are proper and for α > 1 the result is sharp.
Theorem 2.2.
Theorem 2.2. A sufficient condition for a function of the form (1) to be in the ME(α) is that ∞ X n=0 [1 + α(n + 1)] |an| ≤1.
Theorem 2.2. A sufficient condition for a function of the form (1) to be in the ME(α) is that ∞ X n=0 [1 + α(n + 1)] |an| ≤1.
Lemma 1.
Lemma 1. Let a function w(z) = a + wmzm + · · · be analytic in the unit disc with w(z) ̸= a and m ≥1. If z0 = r0eiθ (0 < r0 < 1) and…
Lemma 1. Let a function w(z) = a + wmzm + · · · be analytic in the unit disc with w(z) ̸= a and m ≥1. If z0 = r0eiθ (0 < r0 < 1) and |w(z0)| = max|z|≤r0 |w(z)|. Then z0w′(z0) = kw(z0) and ℜ 1 + z0w′′(z0) w′(z0) ≥k, where k is real and k ≥m.
Theorem 2.3.
Theorem 2.3. If the function f given by (1) belongs to the class ME(α), then |an| ≤ 2 p α2(n + 1)2 + 1 + α(n + 1), n ≥0. (6) The result is…
Theorem 2.3. If the function f given by (1) belongs to the class ME(α), then |an| ≤ 2 p α2(n + 1)2 + 1 + α(n + 1) , n ≥0. (6) The result is sharp for function zf(z) = 1+dnzn 1−dnzn where dn = √ α2n2 + 1 + αn.
Theorem 3.1
Theorem 3.1 f ∈ME(α) ⇔Rez(f(z) ∗1+z(αeiγ−1) z(1−z)2 ) > 0, γ ∈(−π, π], z ∈E.
Theorem 3.1 f ∈ME(α) ⇔Rez(f(z) ∗1+z(αeiγ−1) z(1−z)2 ) > 0, γ ∈(−π, π], z ∈E.
Theorem 3.2.
Theorem 3.2. If f(z)−ǫz−1 1−ǫ ∈ME(α), for δ < ǫ < 1, then Nγ(f) ⊂ME(α) where γ = 1 1+2α.
Theorem 3.2. If f(z)−ǫz−1 1−ǫ ∈ME(α), for δ < ǫ < 1, then Nγ(f) ⊂ME(α) where γ = 1 1+2α.
Theorem 4.1.
Theorem 4.1. A function f of the form f(z) = z−1−P∞ n=1 anzn is in T ME(α) if and only if ∞ X n=1 (1 + α(n + 1))an ≤1. The result is sharp…
Theorem 4.1. A function f of the form f(z) = z−1−P∞ n=1 anzn is in T ME(α) if and only if ∞ X n=1 (1 + α(n + 1))an ≤1. The result is sharp for the function f(z) given by f(z) = z−1 −( 1 1 + α(n + 1))zn, n = 1, 2, 3, ....
Corollary 1.
Corollary 1. The extreme points of T ME(α) are f1(z) = z−1 and fn(z) = z−1 − zn 1 + α(n + 1), n = 1, 2, 3,....
Corollary 1. The extreme points of T ME(α) are f1(z) = z−1 and fn(z) = z−1 − zn 1 + α(n + 1), n = 1, 2, 3, ... .
Corollary 2
Corollary 2. If f(z) = z−1 −P∞ n=1 anzn, an ≥0 is in T ME(α), then 1 r − 1 1 + 2αr ≤|f(z)| ≤1 r + 1 1 + 2αr, with equality for f(z) = 1 z −…
Corollary 2 . If f(z) = z−1 −P∞ n=1 anzn, an ≥0 is in T ME(α), then 1 r − 1 1 + 2αr ≤|f(z)| ≤1 r + 1 1 + 2αr, with equality for f(z) = 1 z − 1 1+2αz at z = r, ir. Finally we prove
Theorem 4.2
Theorem 4.2. Let f ∈P be given by (1) and define the partial sums S1(z) and Sn(z) by S1(z) = z−1 and Sn(z) = z−1 + Pn−1 k=1 akzk. Suppose…
Theorem 4.2 . Let f ∈P be given by (1) and define the partial sums S1(z) and Sn(z) by S1(z) = z−1 and Sn(z) = z−1 + Pn−1 k=1 akzk. Suppose also that ∞ X K=1 dk|ak| ≤1 (dk = 1 + α(n + 1)). (11) Then we have Re( f(z) Sn(z)) > 1−1 dn and
Function classes studied:
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