Abstract
For functions of the form f(z) = zp + P∞
n=1 ap+nzp+n we obtain
sharp bounds for some coefficients functionals in certain subclasses of starlike
functions. Certain applications of our main results are also given. In par-
ticular, Fekete–Szeg¨o-like inequality for classes of functions defined through
extended fractional differintegrals are obtained.
Results & Lemmas (7)
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Lemma 1.2
Lemma 1.2 ([1]). If w ∈Ω, then |w2 −tw2 1| ≤ −t if t ≤−1, 1 if −1 ≤t ≤1, t if t ≥1.
Lemma 1.2 ([1]). If w ∈Ω, then |w2 −tw2 1| ≤ −t if t ≤−1, 1 if −1 ≤t ≤1, t if t ≥1.
Lemma 1.2
Lemma 1.2 is a reformulation of Lemma of Ma and Minda [4].
Lemma 1.2 is a reformulation of Lemma of Ma and Minda [4].
Lemma 1.3
Lemma 1.3 ([3]). If w ∈Ω, then for any complex number t |w2 −tw2 1| ≤max 1, |t|. The result is sharp for the functions w(z) = z or w(z) =…
Lemma 1.3 ([3]). If w ∈Ω, then for any complex number t |w2 −tw2 1| ≤max{1, |t|}. The result is sharp for the functions w(z) = z or w(z) = z2.
Lemma 1.4
Lemma 1.4 ([8]). If w ∈Ω, then for any real numbers q1 and q2 the following sharp estimate holds: |w3 + q1w1w2 + q2w3 1| ≤H(q1, q2) where…
Lemma 1.4 ([8]). If w ∈Ω, then for any real numbers q1 and q2 the following sharp estimate holds: |w3 + q1w1w2 + q2w3 1| ≤H(q1, q2) where H(q1, q2)=
Theorem 2.1.
Theorem 2.1. Let φ(z) = 1 + B1z + B2z2 +..., where Bn’s are real with B1 > 0 and B2 ≥0, let 0 < λ ≤1, and σ1:= [pB2 1λ + (B2 −B1)(p −pλ +…
Theorem 2.1. Let φ(z) = 1 + B1z + B2z2 + . . . , where Bn’s are real with B1 > 0 and B2 ≥0, let 0 < λ ≤1, and σ1 := [pB2 1λ + (B2 −B1)(p −pλ + 1)](p −pλ + 1) (p −pλ + 2)pB2 1 , σ2 := [pB2 1λ + (B2 + B1)(p −pλ + 1)](p −pλ + 1) (p −pλ + 2)pB2 1 , σ3 := [pB2 1λ + B2(p −pλ + 1)](p −pλ + 1) (p −pλ + 2)pB2
Corollary 2.5.
Corollary 2.5. Let φ(z) be as in Theorem 2.1, g(z) = zp + ∞ X n=1 gp+nzp+n (gp+n > 0), and let σ1:= g2 p+1 gp+2 [pB2 1λ + (B2 −B1)(p −pλ +…
Corollary 2.5. Let φ(z) be as in Theorem 2.1, g(z) = zp + ∞ X n=1 gp+nzp+n (gp+n > 0), and let σ1 := g2 p+1 gp+2 [pB2 1λ + (B2 −B1)(p −pλ + 1)](p −pλ + 1) (p −pλ + 2)pB2 1 ,
Theorem 3.4.
Theorem 3.4. Let φ(z) be as in Theorem 2.1, and let σ1:= (p + 1)(p + 2 −δ) (p + 2)(p + 1 −δ) [pB2 1λ + (B2 −B1)(p −pλ + 1)](p −pλ + 1) (p…
Theorem 3.4. Let φ(z) be as in Theorem 2.1, and let σ1 := (p + 1)(p + 2 −δ) (p + 2)(p + 1 −δ) [pB2 1λ + (B2 −B1)(p −pλ + 1)](p −pλ + 1) (p −pλ + 2)pB2 1 , σ2 := (p + 1)(p + 2 −δ) (p + 2)(p + 1 −δ) [pB2 1λ + (B2 + B1)(p −pλ + 1)](p −pλ + 1) (p −pλ + 2)pB2 1 ,
Function classes studied:
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