Results & Lemmas (3)
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Theorem 1.
Theorem 1. &v c. *, for all real y.
Theorem 1. &v c. $*, for all real y.$
Theorem 2.
Theorem 2. If 0 < <5 < y (or y < 6 < 0) then f£ycz&„.
Theorem 2. If 0 < <5 < y (or y < 6 < 0) then f£ycz&„.
Theorem 3.
Theorem 3. If fe IPV, f(z) — z ~ a.,z2 + n, and if p is a complex constant, then (12) [a2(l+y)|<2, (13) |a3(l+8y) + al(/--ly-2)|<4, (31) (1…
Theorem 3. If fe IPV, f(z) — z ~ a.,z2 + n, and if p is a complex constant, then (12) [a2(l+y)|<2, (13) |a3(l+8y) + al(/--ly-2)|<4, (31) (1 + y)’|l+2y11«3-/za2|< Max[(1 + y)«, |4,u(l+2y)-3(3y+1)|] all hold. Remarks, (i) y = 0 (14) reduces to |a8 —^«£1 < Maxfl, |4/4-3|], which is a result of Keogh and Merkes [2]. (ii) For y — 1 (Id) reduces to (iii) Iîf(z)e^CY, y > 0, then (15) |oa|<2/(3+y)
Definitions (2)
Def 1.
Definition 1. Let /(z) = z + £ anzn be regular in the unit disc D, with /(z), f'(z and [1+»/"(»)//'(«)] =#= 0 in 0 < |z| < 1. Suppose у is…
Definition 1. Let /(z) = z + £ anzn be regular in the unit disc D, with /(z), f'(z} and [1+»/"(»)//'(«)] =#= 0 in 0 < |z| < 1. Suppose у is real and (1) for zeD, where the powers appearing in (1) are meant as principal values. 1 This work was carried out while the second author was an IREX Scholar in Poland.
Def 2.
Definition 2. Let f(z) = z+ anzn be regular in the unit disc D with 2 f(z),f'(z), l+zf"(z)/f'(z) ^0 in 0 < |»| < 1, and suppose y is a real…
Definition 2. Let f(z) = z+\ anzn be regular in the unit disc D with 2 f(z),f'(z), l+zf"(z)/f'(z) ^0 in 0 < |»| < 1, and suppose y is a real con stant, 0 < y < 3. If for 2+, then we say that /(«) is a gamma-starlike function of order a, and we denote the class of such functions by 3y{a). If (18) is replaced by for ze D, then we say that f(z) is a strongly gamma-starlike function of order a, and we denote the class of such functions by 3y (a). Note that 3*(a) and 3*(a) are respectively the class