Narrow Results

This paper looks at subordination properties involving 1 + βzp′(z) and the class of Sokół–Stankiewicz starlike functions where condition on β is determined so that the relation would imply . Similar results are obtained for expressions of the form and . These results are then applied to obtain other relations involving similar expressions.

The $k$-convoluted operators related to the $k$-Whittaker function, confluent hypergeometric function of the first kind, have been developed using the $k$-symbol calculus in which this sort of calculus presents a generalization of the gamma function. $K$-symbol fractional calculus is employed to generalize and extend many differential and integral operators of fractional calculus. Based on this premise, a new geometric formula for normalized functions in the symmetric domain known as the open unit disk using the conformable fractional differential operator has been presented in this study. Thus, our technique entails investigating the most well-known geometric properties of this new operator, such as the subordination features and coefficient bounds so that the theory of differential subordination can be adjusted accordingly. By means of this technique, numerical results have been investigated for the proposed method. To this end, a few prominent corollaries of our primary findings as standout instances have been pointed out based on the positivity of the solutions, computational and numerical analyses.

Non-local operators of differentiation are bestowed with capabilities of encompassing complex natural into mathematical equations. Symmetry as invariance under a specified group of transformations can allow for the concept to be applied extensively not only to spatial figures but also to abstract objects like mathematical expressions which can be said to be expressions of physical relevance, in particular dynamical equations. Derived from this point of view, it can be noted that the more complex physical problems are, the more complex mathematical operators of differentiation are required. Accordingly, the fractal–fractional operators (FFOs) are expanded into the complex plane in our research which revolves around a unique class of normalized analytic functions in the open unit disk. To bring FFOs (differential and integral) into the normalized class, the study aims to expand and modify them along with the investigation of the FFOs geometrically. The qualities of convexity and starlikeness are implicated in this study where the differential subordination technique serves as the foundation for the inquiry under consideration. Furthermore, a collection of differential FFO inequalities is taken into account, demonstrating that the normalized Fox–Wright function can contain all FFOs. Besides these steps, the concept of Grunsky factors is applied to investigate symmetry, while boundary value issues involving FFOs are probed. Consequently, the related properties and applications can be further developed, which requires the devotion to differential fractional problems and diverse complex problems in relation to viable applications, pointing out the room to modify and upgrade the existing methods for more optimal outcomes in challenging real-world problems.

We consider a two-dimensional Hele–Shaw flow in a wedge under injection into wedge's vertex. An isoperimetric inequality is obtained. It shows that the rate of the area growth is controlled by a special conformal quantity-triangle conformal radius.

In this work, we introduce some properties of a complex fractional differential operator. The boundedness and some other properties are studied in view of geometric function theory.

In this paper, we define a new integral operator in the open unit disk. This operator is considered as a complex Volterra operator. Moreover, we define a new subspace of Hardy space involving the normalized analytic functions. We shall show that the new integral operator is closed in the subspace of normalized functions. Geometric characterizations are established in the sequel. Our display is maintained by the Jack Lemma.

A typical quandary in geometric functions theory is to study a functional composed of amalgamations of the coefficients of the pristine function. Conventionally, there is a parameter over which the extremal value of the functional is needed. The present paper deals with consequential functional of this type. By making use of Hohlov operator, a new subclass ${{ℛ}}_{a,b}^c$ of analytic functions defined ins the open unit disk is introduced. For both real and complex parameter, the sharp bounds for the Fekete–Szegö problems are found. An attempt has also been taken to found the sharp upper bound to the second and third Hankel determinant for functions belonging to this class. All the extremal functions are express in term of Gauss hypergeometric function and convolution. Finally, the sufficient condition for functions to be in ${{ℛ}}_{a,b}^c$ is derived. Relevant connections of the new results with well-known ones are pointed out.

In this investigation, we introduce and study two new subclasses of bi-univalent functions defined by using the function $f_{\delta }$ and Salagean differential operator. Furthermore, we find estimates on the coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for these function classes.

We present the sharp bounds for the third Hankel determinant $|H_{3,1}(f^{-1})|$ and Zalcman functional $|t_3^2-t_5|$ of the inverse function of the familiar subfamily of starlike functions associated with the right half of lemniscate of Bernoulli.

We consider the class of $\lambda$-pseudo starlike functions $f(z)=z+a_2{z}^2+a_3{z}^3+\cdots$ such that $z{(f^{\prime }(z))}^{\lambda }/f(z)$ maps the open unit disk $\left|z\right|<1$ onto a strip domain $w$ with $\frac{\alpha -\pi }{2sin\alpha }<\Re (w(z))<\frac{\alpha }{2sin\alpha }$ for some $\alpha$, $\pi /2\le \alpha <\pi$. We estimate $\left|a_2\right|$, $\left|a_3\right|$ and solve the Fekete–Szegö problem for functions in this class.

In this paper, we introduce and study a new subclass of starlike functions with respect to $(2ȷ,\ell )$-symmetric conjugate points using the principle of subordination. Several relationship with the well-known classes have been established. We have focussed on conic regions when it pertained to applications of our main results. Inclusion results, subordination property and coefficient inequality of the defined class are the main results of this paper. The applications of the results are presented here as corollaries, most of which are extensions of well-known results.

In this paper, we use properties of the hyperbolic metric and the modular function to show that Bohr’s radius for covering maps onto hyperbolic domains is $\ge e^{-\pi }.$ This includes almost all known classes of analytic functions.

The Grunsky and Teichmüller norms ϰ(f) and k(f) of a univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to are related by ϰ(f) ≤ k(f). In 1985, Jürgen Moser conjectured that any univalent function in the disk Δ^* = {z : |z| > 1} can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kühnau and the author. We prove that approximation is possible in a much stronger sense and also give applications. One of them is the inversion of Ahlfors and Grunsky inequalities, which solves the old Kühnau problem on exact lower bound in the inverse inequality estimating k(f) by ϰ(f).