Abstract
We have constructed a subclass of analytic bi-univalent functions using (p,q)-Lucas
polynomials in this research contribution. Bounds for certain coefficients and
Fekete–Szego¨ inequalities have been estimated.
Results & Lemmas (7)
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Theorem 1.1
Theorem 1.1 (see [1, 8]) Let GLnðxÞðzÞ be the generating function of the ðp; qÞ- Lucas polynomial sequence Lp;q;nðxÞ. Then: 123 1016 S.…
Theorem 1.1 (see [1, 8]) Let GLnðxÞðzÞ be the generating function of the ðp; qÞ- Lucas polynomial sequence Lp;q;nðxÞ. Then: 123 1016 S. Yalçın et al.
Theorem 2.1
Theorem 2.1 Let f represented by (1.1) belong to the class ARð1; t; xÞ. Then: ja2j jpðxÞj ffiffiffiffiffiffiffiffiffiffiffi jpðxÞj p…
Theorem 2.1 Let f represented by (1.1) belong to the class ARð1; t; xÞ. Then: ja2j jpðxÞj ffiffiffiffiffiffiffiffiffiffiffi jpðxÞj p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j ð1 þ tÞ2 þ 2512t2 þ 10ð1 þ tÞ1t p2ðxÞ þ 2 1 þ ð1 þ tÞ2 þ 2512t2 þ 2ð1 þ tÞ þ 101t þ 10ð1 þ tÞ1t qðxÞj
Corollary 2.1
Corollary 2.1 Let f 2 DRð1; xÞ. Then: ja2j jpðxÞj ffiffiffiffiffiffiffiffiffiffiffi jpðxÞj p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j12p2ðxÞ þ 2ð1 þ…
Corollary 2.1 Let f 2 DRð1; xÞ. Then: ja2j jpðxÞj ffiffiffiffiffiffiffiffiffiffiffi jpðxÞj p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j12p2ðxÞ þ 2ð1 þ 1Þ2qðxÞ q j : ja3j p2ðxÞ ð1 þ 1Þ2 þ jpðxÞj
Corollary 2.2
Corollary 2.2 Let f 2 HRðxÞ. Then: ja2j jpðxÞj ffiffiffiffiffiffiffiffiffiffiffi jpðxÞj p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jp2ðxÞ þ 8qðxÞj p ja3j p2ðxÞ 4 þ…
Corollary 2.2 Let f 2 HRðxÞ. Then: ja2j jpðxÞj ffiffiffiffiffiffiffiffiffiffiffi jpðxÞj p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jp2ðxÞ þ 8qðxÞj p ja3j p2ðxÞ 4 þ jpðxÞj 3 :
Theorem 3.1
Theorem 3.1 Let f 2 ARð1; t; xÞ and g 2 R. Then: ja3 ga2 2j jpðxÞj 1 þ 2ð1 þ tÞ þ 101t; jg 1j j1 ð1þð1þtÞþ51tÞ2ðp2ðxÞþ2qðxÞÞ…
Theorem 3.1 Let f 2 ARð1; t; xÞ and g 2 R. Then: ja3 ga2 2j jpðxÞj 1 þ 2ð1 þ tÞ þ 101t ; jg 1j j1 ð1þð1þtÞþ51tÞ2ðp2ðxÞþ2qðxÞÞ ð1þ2ð1þtÞþ101tÞp2ðxÞ j: jp3ðxÞjjg1j j ð1þ1Þ2þ2512t2þ10ð1þtÞ1t ð Þp2ðxÞþ2qðxÞ 1þð1þtÞþ51t ð Þ2j;
Corollary 3.1
Corollary 3.1 Let f 2 DRð1; xÞ and g 2 R. Then: ja3 ga2 2j jpðxÞj 1 þ 21; jg 1j j1 ð1þ1Þ2ðp2ðxÞþ2qðxÞÞ ð1þ21Þp2ðxÞ j:…
Corollary 3.1 Let f 2 DRð1; xÞ and g 2 R. Then: ja3 ga2 2j jpðxÞj 1 þ 21 ; jg 1j j1 ð1þ1Þ2ðp2ðxÞþ2qðxÞÞ ð1þ21Þp2ðxÞ j: jp3ðxÞjjg1j jð1þ1Þ2p2ðxÞþ2qðxÞ 1þ1 ð Þ2j; jg 1j j1 ð1þ1Þ2ðp2ðxÞþ2qðxÞÞ ð1þ21Þp2ðxÞ
Corollary 3.2
Corollary 3.2 Let f 2 HRð1; xÞ and g 2 R. Then: ja3 ga2 2j jpðxÞj 3; jg 1j j1 4ðp2ðxÞþ2qðxÞÞ 3p2ðxÞ j: jp3ðxÞjjg1j…
Corollary 3.2 Let f 2 HRð1; xÞ and g 2 R. Then: ja3 ga2 2j jpðxÞj 3 ; jg 1j j1 4ðp2ðxÞþ2qðxÞÞ 3p2ðxÞ j: jp3ðxÞjjg1j 4jp2ðxÞþ2qðxÞj; jg 1j j1 4ðp2ðxÞþ2qðxÞÞ 3p2ðxÞ j:
Definitions (2)
Def 1.1
Definition 1.1 (see [1, 8]) Consider two polynomials pðxÞ and qðxÞ with real coefficients. The ðp; qÞ-Lucas polynomial Lp;q;nðxÞ is given by…
Definition 1.1 (see [1, 8]) Consider two polynomials pðxÞ and qðxÞ with real coefficients. The ðp; qÞ-Lucas polynomial Lp;q;nðxÞ is given by the recurrence relation: Lp;q;nðxÞ ¼ pðxÞLp;q;n1ðxÞ þ qðxÞLp;q;n2ðxÞ ðn 2Þ: The first few Lucas polynomials are listed below: Lp;q;0ðxÞ ¼ 2; Lp;q;1ðxÞ ¼ pðxÞ; Lp;q;2ðxÞ ¼ p2ðxÞ þ 2qðxÞ; Lp;q;3ðxÞ ¼ p3ðxÞ þ 3pðxÞqðxÞ; ::: ð1:2Þ
Def 1.2
Definition 1.2 A function f 2 R is said to be in the class: ARð1; t; xÞ ð1 0; 0 t 1; z; w 2 UÞ if ð1 1Þð1 tÞ fðzÞ z þ ðt þ 1ð1 þ…
Definition 1.2 A function f 2 R is said to be in the class: ARð1; t; xÞ ð1 0; 0 t 1; z; w 2 UÞ if ð1 1Þð1 tÞ fðzÞ z þ ðt þ 1ð1 þ tÞÞf0ðzÞ þ 1tðzf00ðzÞ 2Þ 1 þ GLp;q;nðxÞðzÞ; and ð1 1Þð1 tÞ gðwÞ w
Function classes studied:
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