PARTIAL SUMS OF FUNCTIONS OF BOUNDED TURNING

We determine conditions under which the partial sums of the Libera integral operator of functions of bounded turning are also of bounded turning.

For f of the form (1.1), the Libera integral operator F is given by The nth partial sums F n (z) of the Libera integral operator F(z) are given by In [6] it was shown that if f ∈ Ꮽ is starlike of order α, α = 0.294,..., so is the Libera integral operator F .We also know that (see, e.g., [1]) there are functions which are univalent or spiral-like in ᐁ so that their Libera integral operators are not univalent or spiral-like in ᐁ.Li and Owa [5] proved that if f ∈ Ꮽ is univalent in ᐁ, then F n (z) is starlike in |z| < 3/8.The number 3/8 is sharp.In this note we make use of a result of Gasper [2] to provide a simple proof for the following theorem.

Preliminary lemmas.
To prove our Main theorem, we will need the following three lemmas.The first lemma is due to Gasper (see [2,Theorem 1]) and the third lemma is a well-known and celebrated result (cf.[3]) that can be derived from the Herglotz' representation for positive real part functions.Lemma 2.1.Let θ be a real number and let m and k be natural numbers.Then Proof.For 0 ≤ r < 1 and for 0 ≤ |θ| ≤ π , write z = r e iθ = r (cos(θ) + i sin(θ)).By DeMoivre's law and the minimum principle for harmonic functions, we have Now by Abel's lemma (cf.Titchmarsh [7]) and condition (2.1) of Lemma 2.1 we conclude that the right-hand side of (2.3) is greater than or equal to −1/3.

Proof of Main theorem.
Let f be of the form (1.1) and belong to Ꮾ(α) for 1/4 ≤ α < 1.Since (f (z)) > α, we have Applying the convolution properties of power series to F n (z), we may write Applying a simple algebra to inequality (3.3) and Q(z) in (3.2) yields On the other hand, the power series P (z) in (3.2) in conjunction with the condition (3.1) yield (P (z)) > 1/2.Therefore, by Lemma 2.3, (F n (z)) > (4α − 1)/3.This concludes the Main theorem.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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1 .
Introduction.Let Ꮽ denote the family of functions f which are analytic in the open unit disk ᐁ = {z : |z| < 1} and are normalized by

Lemma 2 . 3 .
Let P (z) be analytic in ᐁ, P (0) = 1 and let (P (z)) > 1/2 in ᐁ.For functions Q analytic in ᐁ, the convolution function P * Q takes values in the convex hull of the image on ᐁ under Q.The operator " * " stands for the Hadamard product or convolution of two power series f (z) = ∞ k=1 a k z k and g(z) = ∞ k=1 b k z k denoted by (f * g)(z) = ∞ k=1 a k b k z k .