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Abstract
The main object of this paper is to study Fekete-Szeg¨o
problem for a certain subclass of p - valent analytic functions.
Fekete-Szeg¨o inequality of several classes are obtained as special
cases from our results. Applications of the result are also obtained
on the class defined by convolution.
AMS Subject Classification (2000). 30C45.
Results & Lemmas (13)
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.
Lemma 2.1.
Lemma 2.1. [12] Let p(z) = 1 + c1z + c2z2 + · · · is analytic function with positive real part in U and v is complex number, then |c2 −vc2…
Lemma 2.1. [12] Let p(z) = 1 + c1z + c2z2 + · · · is analytic function with positive real part in U and v is complex number, then |c2 −vc2 1| ≤2 max {1, |2v −1|} , the result is sharp for functions given by p(z) = 1 + z2 1 −z2, p(z) = 1 + z 1 −z.
Lemma 2.2.
Lemma 2.2. [12] Let p(z) = 1 + c1z + c2z2 + · · · is analytic function with positive real part in U,then |c2 −vc2 1| ≤ −4v + 2,…
Lemma 2.2. [12] Let p(z) = 1 + c1z + c2z2 + · · · is analytic function with positive real part in U,then |c2 −vc2 1| ≤ −4v + 2, if v ≤0; 2, if 0 ≤v ≤1;
Theorem 2.3.
Theorem 2.3. Let φ(z) = 1 + B1z + B2z2 + · · · (B1 ̸= 0). If f(z) ∈Cn,γ p,q (φ), then |ap+2 −µa2 p+1| ≤|γ| (p −q + 1)(p −q)n (p −q +…
Theorem 2.3. Let φ(z) = 1 + B1z + B2z2 + · · · (B1 ̸= 0). If f(z) ∈Cn,γ p,q (φ), then |ap+2 −µa2 p+1| ≤|γ| (p −q + 1)(p −q)n (p −q + 2)n−1(p + 1)(p + 2) |B1| 2 max 1; B2 B1 + 2k1
Corollary 2.4.
Corollary 2.4. Let f(z) ∈A satisfy the inequality α < Re 1 + 1 γ zf ′(z) f(z) −1 < β, (2.6) then |a3−µa2 2| ≤|γ|(β −α) √ 2π
Corollary 2.4. Let f(z) ∈A satisfy the inequality α < Re 1 + 1 γ zf ′(z) f(z) −1 < β, (2.6) then |a3−µa2 2| ≤|γ|(β −α) √ 2π
Corollary 2.5.
Corollary 2.5. Let f(z) ∈A satisfy the inequality α < Re 1 + 1 γ zf ′′(z) f(z) < β, then |a3 −µa2 2| ≤ |γ|(β −α) 3 √
Corollary 2.5. Let f(z) ∈A satisfy the inequality α < Re 1 + 1 γ zf ′′(z) f(z) < β, then |a3 −µa2 2| ≤ |γ|(β −α) 3 √
Corollary 2.6.
Corollary 2.6. [19] Let φ(z) = 1 + B1z + B2z2 + · · · with B1 ̸= 0). If f(z) ∈Sγ(φ), then |a3 −µa2 2| ≤|γ||B1| 2 max 1,
Corollary 2.6. [19] Let φ(z) = 1 + B1z + B2z2 + · · · with B1 ̸= 0). If f(z) ∈Sγ(φ), then |a3 −µa2 2| ≤|γ||B1| 2 max 1,
Corollary 2.7.
Corollary 2.7. Let φ(z) = 1 + B1z + B2z2 + · · · with B1 ̸= 0). If f(z) ∈ Cγ(φ), then |a3 −µa2 2| ≤|γ||B1| 6 max 1,
Corollary 2.7. Let φ(z) = 1 + B1z + B2z2 + · · · with B1 ̸= 0). If f(z) ∈ Cγ(φ), then |a3 −µa2 2| ≤|γ||B1| 6 max 1,
Theorem 2.8.
Theorem 2.8. Let φ(z) = 1 + B1z + B2z2 + · · · (B1 > 0). If f(z) given by (1.3) belongs to Cn,γ p,q (φ), then |ap+2 −µa2 p+1| ≤ …
Theorem 2.8. Let φ(z) = 1 + B1z + B2z2 + · · · (B1 > 0). If f(z) given by (1.3) belongs to Cn,γ p,q (φ), then |ap+2 −µa2 p+1| ≤ 2γk1 [B2 1 + B2 −µγB2 1k2] , if µ ≤σ1;
Theorem 2.9.
Theorem 2.9. Let φ(z) = 1 + B1z + B2z2 + · · · (B1 > 0). If f(z) given by(1.3) belongs to Cn,γ p,q (φ) and σ3 is given by σ3 = 1 γB1k2 B1…
Theorem 2.9. Let φ(z) = 1 + B1z + B2z2 + · · · (B1 > 0). If f(z) given by(1.3) belongs to Cn,γ p,q (φ) and σ3 is given by σ3 = 1 γB1k2 B1 + B2 B1 . If σ1 < µ ≤σ3, then |ap+2 −µa2 p+1| + 1
Corollary 2.10.
Corollary 2.10. (see [24]) Let φ(z) = 1+B1z +B2z2 +· · · with B1 > 0 and B2 ≥0. Let σ4 = γB2 1 + (B2 −B1) 2γB2 1, σ5 = γB2 1 + (B2 + B1)…
Corollary 2.10. (see [24]) Let φ(z) = 1+B1z +B2z2 +· · · with B1 > 0 and B2 ≥0. Let σ4 = γB2 1 + (B2 −B1) 2γB2 1 , σ5 = γB2 1 + (B2 + B1) 2γB2 1 , σ6 = γB2 1 + B2 2γB2
Corollary 2.11.
Corollary 2.11. (see [24]) Let φ(z) = 1+B1z +B2z2 +· · · with B1 > 0 and B2 ≥0. Let χ1 = 2[γB2 1 + B2 −B1] γB2 1, χ2 = 2[γB2 1 + B2 + B1]…
Corollary 2.11. (see [24]) Let φ(z) = 1+B1z +B2z2 +· · · with B1 > 0 and B2 ≥0. Let χ1 = 2[γB2 1 + B2 −B1] γB2 1 , χ2 = 2[γB2 1 + B2 + B1] γB2 1 , χ3 = 2[γB2 1 + B2] γB2
Theorem 3.1.
Theorem 3.1. Let g(z) = 1+g1z +g2z2 +· · · (gn > 0). If f(z) given by(1.3) belongs to Cn,γ p,q (φ), then |ap+2 −µa2 p+1| ≤ …
Theorem 3.1. Let g(z) = 1+g1z +g2z2 +· · · (gn > 0). If f(z) given by(1.3) belongs to Cn,γ p,q (φ), then |ap+2 −µa2 p+1| ≤ 2γk1 gp+2 h
Theorem 3.2.
Theorem 3.2. Let φ(z) = 1 + B1z + B2z2 + · · ·, where Bn’s are real with B1 > 0 and B2 ≥0. If f(z) given by(1.3) belongs to Cn,γ p,q (φ),…
Theorem 3.2. Let φ(z) = 1 + B1z + B2z2 + · · · , where Bn’s are real with B1 > 0 and B2 ≥0. If f(z) given by(1.3) belongs to Cn,γ p,q (φ), then |ap+2 −µa2 p+1| ≤ (p+1−δ)(p+2−δ) (p+1)(p+2) 2γk1
Function classes studied:
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