Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
Abstract
The objective of this paper is to introduce new classes of m-fold symmetric
bi-univalent functions. We discuss estimates on the Taylor–Maclaurin coefficients
|am+1| and |a2m+1|, and the Fekete–Szeg˝o problem is also considered for the new
classes of functions introduced. We denote these classes by MF – Sp,q
,m(h), MF – Sp,q
,m(s),
and MF – Sb,d
,m. Quantum calculus aspects are also considered in this study to
enhance its novelty and to obtain more interesting results.
MSC: 30C45; 30C50
Results & Lemmas (6)
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 4
Lemma 4 [14, 29] Let the function w ∈P be given by the following series w(z) = 1 + w1z + w2z2 +···, z ∈U, where we denote by P the class of…
Lemma 4 [14, 29] Let the function w ∈P be given by the following series w(z) = 1 + w1z + w2z2 +··· , z ∈U, where we denote by P the class of Carathéodory functions analytic in the open disk U, P =
Theorem 6
Theorem 6 If the function f, given by relation (3), is in the function class MF –Sp,q ,m(h),(m ∈ N,0 < q < p ≤1,0 < h ≤1,(z,w) ∈U), then…
Theorem 6 If the function f , given by relation (3), is in the function class MF –Sp,q ,m(h),(m ∈ N,0 < q < p ≤1,0 < h ≤1,(z,w) ∈U), then the following inequalities are true: |am+1| ≤ 2h (m + 1)h[2m + 1]p,q(1 + 2m) – (h – 1)(1 + m)2[m + 1]2p,q (10) and |a2m+1| ≤ 2h (1 + 2m)[2m + 1]p,q + 2h2 (1 + m)[m + 1]2p,q
Theorem 7
Theorem 7 Let f be a function of the form (3) in the class MF – Sp,q ,m(h). Then a2m+1 – ρa2 m+1 ≤ ⎧ ⎨ ⎩ 2h (1+2m)[2m+1]p,q, |l(ρ)| ≤…
Theorem 7 Let f be a function of the form (3) in the class MF – Sp,q ,m(h). Then a2m+1 – ρa2 m+1 ≤ ⎧ ⎨ ⎩ 2h (1+2m)[2m+1]p,q , |l(ρ)| ≤ 1 (1+2m)[2m+1]p,q , 4h(1 + 2m)[2m + 1]2 p,q|l(ρ)|,
Theorem 9
Theorem 9 Let f be a function in the class MF – Sp,q ,m(s),(m ∈N,0 < q < p ≤1,0 ≤s < 1,(z,w) ∈U), which has the form (3). Then |am+1| ≤min
Theorem 9 Let f be a function in the class MF – Sp,q ,m(s),(m ∈N,0 < q < p ≤1,0 ≤s < 1,(z,w) ∈U), which has the form (3). Then |am+1| ≤min
Theorem 10
Theorem 10 Let f be a function of the form (3) in the class MF – Sp,q ,m(s). Then a2m+1 – ρa2 m+1 ≤ ⎧ ⎨ ⎩ 2(1–s) (1+2m)[2m+1]p,q,…
Theorem 10 Let f be a function of the form (3) in the class MF – Sp,q ,m(s). Then a2m+1 – ρa2 m+1 ≤ ⎧ ⎨ ⎩ 2(1–s) (1+2m)[2m+1]p,q , |l(ρ)| ≤ 1 2(1+2m)[2m+1]p,q , 4(1 + 2m)(1 – s)[2m + 1]2 p,q|l(ρ)|,
Theorem 12
Theorem 12 If the function f of the form (3) is in the class MF – Sb,d ,m, then the following inequalities are satisfied: |am+1| ≤min |b′…
Theorem 12 If the function f of the form (3) is in the class MF – Sb,d ,m, then the following inequalities are satisfied: |am+1| ≤min |b′ 1(0)|2 + |d′ 1(0)|2 2(1 + m)2[m + 1]2p,q , |b′′ 2(0)| + |d′′ 2(0)| (1 + 2m)(m + 1)[2m + 1]p,q
Function classes studied:
Related Papers