Integral means of the Derivatives of some Univalent Functions

“…Again, it suffices to show that F and G satisfy Proposition 6. Condition (i) is trivial, Our Droof shows that the conclusion of Theorem 3 holds for all fe S satisfying Gog I/'D* ^ 0°g I k'\)** m particular for all / in the class B(<x) considered by Leung in [16].…”

Section: Let H{z) = Logfc^z) Ze a Then H Is Univalent And H(a) Is Amentioning

confidence: 91%

Integral Means and Bmoa-Norms of Logarithms of Univalent Functions

Girela

1986

Journal of the London Mathematical Society

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“…As a straightforward adaptations of the known proofs, Miller extended all these three cases to the corresponding subclasses of

scriptS

consisting of m ‐fold convex, starlike and close‐to‐convex functions, respectively. Finally, by making use of the theory of symmetrization developed by Baernstein , Leung extended the result for

F = S^{*}

to the class of Bazilevič functions and the generalized functional

\int_{- π}^{π} Φ (prefixlog | f^{'} ((), r, e^{i θ}) |) d θ

, where Φ is a nondecreasing convex function on

double-struckR

. Recently, the extremal problem for the class of convex functions of order

- 1 / 2

was solved in .…”

Section: Introductionmentioning

confidence: 99%

Circular symmetrization, subordination and arclength problems on convex functions

et al. 2015

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Abstract. We study the class C(Ω) of univalent analytic functions f in the unit disk D = {z ∈ C : |z| < 1} of the form f (z) = z + ∞ n=2 a n z n satisfyingwhere Ω will be a proper subdomain of C which is starlike with respect to 1(∈ Ω). Let φ Ω be the unique conformal mapping of D onto Ω with φ Ω (0) = 1 anddenote the arclength of the image of the circle {z ∈ C : |z| = r}, r ∈ (0, 1). The first result in this paper is an inequality L r (f ) ≤ L r (k Ω ) for f ∈ C(Ω), which solves the general extremal problem max f ∈C(Ω) L r (f ), and contains many other well-known results of the previous authors as special cases. Other results of this article cover another set of related problems about integral means in the general setting of the class C(Ω).

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Integral means of the Derivatives of some Univalent Functions

Cited by 41 publications

References 5 publications

Integral Means of Analytic Mappings by Iteration of Janowski Functions

Integral Means of Analytic Mappings by Iteration of Janowski Functions

Integral Means and Bmoa-Norms of Logarithms of Univalent Functions

Circular symmetrization, subordination and arclength problems on convex functions

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