Abstract
By considering a fixed point in unit disk ∆, a new class of univa-
lent convex functions is defined. Coefficient inequalities, integral operator and
extreme points of this class are obtained.
Results & Lemmas (4)
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Theorem 2.1.
Theorem 2.1. The function f(z) of the form (1.1) belongs t Mw(A, B, m) if and only if +∞ X n=k nm+1[n(B + 1) + A + 2]an ≤1 + A −B. (2.1)
Theorem 2.1. The function f(z) of the form (1.1) belongs t Mw(A, B, m) if and only if +∞ X n=k nm+1[n(B + 1) + A + 2]an ≤1 + A −B. (2.1)
Theorem 2.2.
Theorem 2.2. Let fj(z) defined by fj(z) = 1 z −w + +∞ X n=k an,j(z −w)n, j = 1, 2,... (2.4) be in the class Mw(A, B, m), then the function…
Theorem 2.2. Let fj(z) defined by fj(z) = 1 z −w + +∞ X n=k an,j(z −w)n, j = 1, 2, ... (2.4) be in the class Mw(A, B, m), then the function F(z) = t X j=1 djfj(z) ,
Theorem 3.1.
Theorem 3.1. Let f0(z) = 1 z −w and fn(z) = 1 z −w + 1 + A −B nm+1[n(B + 1) + A + 2](z −w)n, n ≥k (3.1) then f(z) ∈Mw(A, B, m) if and only…
Theorem 3.1. Let f0(z) = 1 z −w and fn(z) = 1 z −w + 1 + A −B nm+1[n(B + 1) + A + 2](z −w)n , n ≥k (3.1) then f(z) ∈Mw(A, B, m) if and only if it can be expressed in the form f(z) = P∞ n=1 Cnfn(z) where Cn ≥0 , Ci = 0 (i = 1, 2, ..., k −1) , P+∞
Theorem 3.2.
Theorem 3.2. Let γ be a real number such that γ > 1. If f(z) ∈Mw(A, B, m), then the functions H1(z) = γ −1 (z −w)γ Z z w (t −w)γ−1f(t)dt…
Theorem 3.2. Let γ be a real number such that γ > 1. If f(z) ∈Mw(A, B, m), then the functions H1(z) = γ −1 (z −w)γ Z z w (t −w)γ−1f(t)dt and H2(z) = C Z 1 0 νCf(ν(z −w) + w)dν , C ≥1 are also in the same class.