Results & Lemmas (18)
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Lemma 1.1.
Lemma 1.1. [20,23] Assume the function r 2 P is expressed by r z r z r z r z z D ( )... ( ), = + + +
Lemma 1.1. [20,23] Assume the function r 2 P is expressed by r z r z r z r z z D ( ) ... ( ), = + + +
Theorem 2.2.
Theorem 2.2. Let a ≥ 0, 0 < q ≤ 1 and l ≥ 0. For every element f in the class q L å (,, ) l a j, defined by (1), we have | | ( ) ( ) (
Theorem 2.2. Let a ≥ 0, 0 < q ≤ 1 and l ≥ 0. For every element f in the class q L å ( , , ) l a j , defined by (1), we have | | ( ) ( ) (
Theorem 2.3.
Theorem 2.3. Let f be a function defined by (1), g w f w ( ) ( ), = -1 0 < q ≤ 1, a ≥ 0 and l ≥ 0 and h 2 C. If f 2 åq(l, a, jL), then we…
Theorem 2.3. Let f be a function defined by (1), g w f w ( ) ( ), = -1 0 < q ≤ 1, a ≥ 0 and l ≥ 0 and h 2 C. If f 2 åq(l, a, jL), then we have a a q q
Theorem 2.6.
Theorem 2.6. Let 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 ≤ m ≤ 1. If the function f is a member of the class åq(l, a, jL), defined by (1), then we…
Theorem 2.6. Let 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 ≤ m ≤ 1. If the function f is a member of the class åq(l, a, jL), defined by (1), then we have | | min , ( ) ( ( )( ) ( a q q
Theorem 2.7.
Theorem 2.7. Let f be a function defined by (1), g w f w ( ) ( ) = -1, 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1 and h 2 C. If f 2 q L m l a j å…
Theorem 2.7. Let f be a function defined by (1), g w f w ( ) ( ) = -1 , 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1 and h 2 C. If f 2 q L m l a j å ( , ,
Corollary 2.8.
Corollary 2.8. Let f be a function defined by (1), g w f w ( ) ( ) = -1, 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 ≤ m ≤ 1. If f 2 q L m l a j å (,,
Corollary 2.8. Let f be a function defined by (1), g w f w ( ) ( ) = -1 , 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 ≤ m ≤ 1. If f 2 q L m l a j å ( , ,
Theorem 3.2.
Theorem 3.2. Let 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 ≤ b ≤ 1. If a function f in the class q L b l a å (,, ) Ã, defined by (1), then we have | |…
Theorem 3.2. Let 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 ≤ b ≤ 1. If a function f in the class q L b l a å ( , , ) Ã , defined by (1), then we have | | ( ) ( )
Corollary 3.3.
Corollary 3.3. Let f be a function defined by (4.1), g w f w ( ) ( ) = -1, 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 ≤ b ≤ 1 and h 2 C. If f 2 q L b l…
Corollary 3.3. Let f be a function defined by (4.1), g w f w ( ) ( ) = -1 , 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 ≤ b ≤ 1 and h 2 C. If f 2 q L b l a å ( , ,
Corollary 3.4.
Corollary 3.4. Let f be a function defined by (1), g w f w ( ) ( ) = -1, 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 ≤ b ≤ 1. If f 2 q L b l a å (,,
Corollary 3.4. Let f be a function defined by (1), g w f w ( ) ( ) = -1 , 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 ≤ b ≤ 1. If f 2 q L b l a å ( , ,
Theorem 3.6.
Theorem 3.6. Let 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 < g < 1. If a function f is a member of the class q L g l a å (,, ), Ã defined by (1), then…
Theorem 3.6. Let 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 < g < 1. If a function f is a member of the class q L g l a å ( , , ), Ã defined by (1), then we have
Corollary 3.7.
Corollary 3.7. Let f be a function defined by (1), g w f w ( ) ( ) = -1, 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 < g < 1 and h 2 C. If f 2 q L g l a å…
Corollary 3.7. Let f be a function defined by (1), g w f w ( ) ( ) = -1 , 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 < g < 1 and h 2 C. If f 2 q L g l a å ( , ,
Corollary 3.8.
Corollary 3.8. Let f be a function defined by (1), g w f w ( ) ( ) = -1, 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 < g < 1. If f 2 q L g l a å (,,
Corollary 3.8. Let f be a function defined by (1), g w f w ( ) ( ) = -1 , 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 < g < 1. If f 2 q L g l a å ( , ,
Theorem 3.10.
Theorem 3.10. Let 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1 and 0 ≤ b ≤ 1. If a function f belongs to the class q L m b l a, (,, ), å Ã defined by…
Theorem 3.10. Let 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1 and 0 ≤ b ≤ 1. If a function f belongs to the class q L m b l a , ( , , ), å Ã defined by (1), then we have
Corollary 3.11.
Corollary 3.11. Let f be a function defined by (1), g w f w ( ) ( ) = -1, 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1, 0 ≤ b ≤ 1 and n 2 C. If f 2 q…
Corollary 3.11. Let f be a function defined by (1), g w f w ( ) ( ) = -1 , 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1, 0 ≤ b ≤ 1 and n 2 C. If f 2 q L m b l a , ( , ,
Corollary 3.12.
Corollary 3.12. Let f be a function defined by (1), g w f w ( ) ( ) = -1, 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1 and 0 ≤ b ≤ 1. If f 2 q L m b…
Corollary 3.12. Let f be a function defined by (1), g w f w ( ) ( ) = -1 , 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1 and 0 ≤ b ≤ 1. If f 2 q L m b l a , ( , ,
Theorem 3.14.
Theorem 3.14. Let 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1 and 0 < g < 1. If a function f belongs to the class q L m g l a, (,, ) å Ã, defined…
Theorem 3.14. Let 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1 and 0 < g < 1. If a function f belongs to the class q L m g l a , ( , , ) å Ã , defined by (1), then we have | | min , ( )( )
Corollary 3.15.
Corollary 3.15. Let f be a function defined by (1), g w f w ( ) ( ) = -1, 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1, 0 < g < 1 n 2 C. If f 2 q L m…
Corollary 3.15. Let f be a function defined by (1), g w f w ( ) ( ) = -1 , 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1, 0 < g < 1 n 2 C. If f 2 q L m g l a , ( , ,
Corollary 3.16.
Corollary 3.16. Let f be a function defined by (1), g w f w ( ) ( ) = -1, 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1 and 0 < g < 1. If f 2 q L m g…
Corollary 3.16. Let f be a function defined by (1), g w f w ( ) ( ) = -1 , 0 < q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1 and 0 < g < 1. If f 2 q L m g l a , ( , ,
Definitions (6)
Def 2.1.
Definition 2.1. Let a l ³ £ £ ³ 0 0 1 0,
Definition 2.1. Let a l ³ £ £ ³ 0 0 1 0 ,
Def 2.5.
Definition 2.5. Let 0 ≤ q ≤ 1, a ≥ 0, l ≥ 0 and 0 ≤ m ≤ 1. A function f 2 å is said to belong in the class q L m l a j å (,, ), if all of…
Definition 2.5. Let 0 ≤ q ≤ 1, a ≥ 0, l ≥ 0 and 0 ≤ m ≤ 1. A function f 2 å is said to belong in the class q L m l a j å ( , , ), if all of the following subordination criteria are met: z f z D f z
Def 3.1.
Definition 3.1. Let 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 ≤ b ≤ 1. A member f 2 å is in the class q L b l a å (,, ) Ã, if each of the following…
Definition 3.1. Let 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 ≤ b ≤ 1. A member f 2 å is in the class q L b l a å ( , , ) Ã , if each of the following conditions (subordination) holds true D
Def 3.5.
Definition 3.5. Let 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 < g < 1. A function f 2 å is in the class q L g l a å (,, ), Ã
Definition 3.5. Let 0 < q ≤ 1, a ≥ 0, l ≥ 0 and 0 < g < 1. A function f 2 å is in the class q L g l a å ( , , ), Ã
Def 3.9.
Definition 3.9. Let 0 ≤ q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1 and 0 ≤ b ≤ 1. A function f 2 å is said to be in the class q L m b l a, (,, ), å Ã…
Definition 3.9. Let 0 ≤ q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1 and 0 ≤ b ≤ 1. A function f 2 å is said to be in the class q L m b l a , ( , , ), å Ã if all of the next subordination conditions are met
Def 3.13.
Definition 3.13. Let 0 ≤ q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1 and 0 < g < 1. A function f 2 å is said to be in the class q L m g l a, (,, ) å Ã,…
Definition 3.13. Let 0 ≤ q ≤ 1, a ≥ 0, l ≥ 0, 0 ≤ m ≤ 1 and 0 < g < 1. A function f 2 å is said to be in the class q L m g l a , ( , , ) å Ã , if each of the following subordination conditions holds true:
Function classes studied:
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