Abstract
Coefficient inequalities, distortion theorem, extreme points and prop
erty preserving integral operators are obtained for certam subclasses of meromor
phic starlike functions with negative coefficints. Convolutions of functions in these
classes are also obtained.
Results & Lemmas (9)
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 1.
Theorem 1. Let J(z) = l/z+ E:=1 an玕be in~. 嶧:=l (n+f3)1anl::; /3-1, then Re(zf'(z)/f(z)) > -(3.
Theorem 1. Let J(z) = l/z+ E:=1 an玕be in~. 嶧:=l (n+f3)1anl::; /3-1, then Re(zf'(z)/f(z)) > -(3.
Theorem 2.
Theorem 2. A function f(z) = ~- 江:=l an玕，an~0 is in EN (/3) if and only if~:=l (n + f3)an~/3 - 1.
Theorem 2. A function f(z) = ~- 江:=l an玕，an~0 is in EN (/3) if and only if~:=l (n + f3)an~/3 - 1.
Theorem 3.
Theorem 3. If f E 霖((3) then 1 3-1 1 /3-1 ''' T l + (3r~IJ(z)I~- + -r T l + (3 (lzl = r) with equality 阮距）＝｝－飪~z (z=ir,r)
Theorem 3. If f E 霖((3) then 1 {3-1 1 /3-1 ''' T l + (3r~IJ(z)I~- + -r T l + (3 (lzl = r) with equality 阮距）＝｝－飪~z (z=ir,r)
Theorem 4.
Theorem 4. If f E 霖 值）then f E 蒿(2 - (3).
Theorem 4. If f E 霖 值）then f E 蒿(2 - (3).
Theorem 5.
Theorem 5. Let f0(z) = l and fn(z) = l - 旦玕or n = z n+P f l, 2,.... Then f E 巧；(/3) if and only if it can be expressed in the form f(z) =…
Theorem 5. Let f0(z) = l and fn(z) = l - 旦玕or n = z n+P f l, 2, . . . . Then f E 巧；(/3) if and only if it can be expressed in the form f(z) = E:'=o -Xnfn(z) where An~0 and E:'=o An = 1.
Theorem 6.
Theorem 6. If f(z) = ~- 霆:=1 anzn, an~0 is in 霖(/3), then F(z) = 寺J訌f(t)dt =~- I::=1 c+~+l an玕，c > 0 belongs to 埡('Y) where'Y =岩躋· The…
Theorem 6. If f(z) = ~- 霆:=1 anzn, an~0 is in 霖(/3), then F(z) = 寺J訌f(t)dt =~- I::=1 c+~+l an玕，c > 0 belongs to 埡('Y) where'Y =岩躋· The result is sharp for f(z) = - - 1 且 z 1 十戶· 00 00
Theorem 7.
Theorem 7. Let F(z) = l/z - 立:=l an玕，an 2: 0 be in Ej.i(/3) and f(z) = ~[(c + l)F(z) + zF'(z)] =~- E:'=1 吐芒玉玕，c > 0. Then Re zf'(z)/f(z) >…
Theorem 7. Let F(z) = l/z - 立:=l an玕，an 2: 0 be in Ej.i(/3) and f(z) = ~[(c + l)F(z) + zF'(z)] =~- E:'=1 吐芒玉玕，c > 0. Then Re zf'(z)/f(z) > -,, (1 < , :'.S 2) for O < lzl < r(扣）. Where r(/3司= inf { (c諡謂隣辶}1/n+l' n = l, 2, .... The result is sharp for F(z) =~- 結玕for some n.
Theorem 8.
Theorem 8. If f(z) = l/z - 立:1 anzn, an~0 and g(z) = 1/z - 立氥bn玕， bn~0 are members of 霹((3), then h(z) = f(z) * g(z) = 1/z - I::=l anbnzz…
Theorem 8. If f(z) = l/z - 立:1 anzn, an~0 and g(z) = 1/z - 立氥bn玕， bn~0 are members of 霹((3), then h(z) = f(z) * g(z) = 1/z - I::=l anbnzz is a member of EN (,) where , = 嗌.!.,
Theorem 9.
Theorem 9. If f(z) E Ej.i(/3) and g(z) E 霖(, ) then f(z) * g(z) E 霖(8) where o =罡
Theorem 9. If f(z) E Ej.i(/3) and g(z) E 霖(, ) then f(z) * g(z) E 霖(8) where o =罡
Function classes studied:
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