Narrow Results

In the present paper, we consider the generalized Hyers–Ulam stability for fractional differential equations of the form:

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.

Let f : (ℝⁿ, 0) → (ℝ, 0) be a nonconstant analytic function defined in a neighborhood of the origin 0 ∈ ℝⁿ. The classical Łojasiewicz inequality states that there exist positive constants δ, c and l such that |f(x)| ≥ cd(x, f^-1(0))^l for ‖x‖ ≤ δ, where d(x, f^-1(0)) denotes the distance from x to the set f^-1(0). The Łojasiewicz exponent of f at the origin 0 ∈ ℝⁿ, denoted by , is the infimum of the exponents l satisfying the Łojasiewicz inequality. In this paper, we establish a formula for computing the Łojasiewicz exponent of f in terms of the Newton polyhedron of f in the case where f is nonnegative and nondegenerate.

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.

In this work, we study some properties of a certain class of meromorphic functions defined by means of the Hadamard product of Cho–Kwon–Srivastava operator. We also define here a similar transformation by means of an operator given by Ghanim and Darus and study its properties.

The objective of this paper is to obtain an upper bound to the second Hankel determinant for the functions belonging to strongly starlike and convex functions of order α(0 < α ≤ 1). Further, we introduce a subclass of analytic functions and obtain the same coefficient inequality for the functions in this class, using Toeplitz determinants.

In this paper, we investigate some strong differential subordination and strong differential superordination results for analytic functions, which involving the iterations of the Owa–Srivastava operator and its combination. Some new sandwich type results are also obtained.

For the class of $\lambda$-bi-starlike functions defined in the unit disc, we find bounds on the modulii of the first three Taylor–Maclaurin’s coefficients. Our estimate on the second and the third coefficients improve upon previously known bounds. The result on the fourth coefficient is new. Similarly, for the $\lambda$-bi-starlike functions defined in the exterior of the unit disc, we find estimates on the first three coefficients. Our findings on the zeroth and the first coefficients extend results known earlier. The results are consummated by refining the first coefficient of a corresponding Carathéodory functions.

In this paper, we introduce the neighborhood ${{𝒩}}_{\delta }(\alpha ,\beta ,\lambda ;g)$ of analytic functions $f$ and $g$ defined in the open unit disc. Furthermore, we derive some sufficient and necessary conditions to be in ${{𝒩}}_{\delta }(\alpha ,\beta ,\lambda ;g)$. In addition, we see some applications of Jack’s lemma.

By using a linear operator associated with the Hurwitz–Lerch zeta function, which is defined here by means of the Hadamard product (or convolution), the authors introduce and investigate certain inclusion relationships of certain subclass of univalent meromorphic functions defined in the punctured unit disc.

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.

In this paper, the following composite analytic functions $F(z)=f(\mathrm{\Phi }(z))$ and $H(z)=G(\mathrm{\Phi }(z),\dots ,\mathrm{\Phi }(z))$ are considered, where $f:ℂ\to ℂ,$$\mathrm{\Phi }:{𝔹}^n\to ℂ,$$G:{ℂ}^m\to ℂ.$ We established conditions which provide equivalence of boundedness of the $l$-index of the function $f$ and boundedness of the $L$-index in joint variables of the function $F,$ where $l:ℂ\to {ℝ}_{+}$ is a continuous function, $L(z)=(l(\mathrm{\Phi }(z))|\frac{\partial \mathrm{\Phi }(z)}{\partial z_1}|,\dots ,$$l(\mathrm{\Phi }(z))\left|\frac{\partial \mathrm{\Phi }(z)}{\partial z_n}\right|).$ For the function $H$ with additional restrictions, the function $L$ is constructed such that $H$ has bounded $L$-index in joint variables in the case when the function $G$ has bounded $L$-index in the direction $1=(1,\dots ,1)$, where $L:{ℂ}^n\to {ℝ}_{+}^n$ is a positive continuous function. Our proofs are based on the application of analog of Hayman’s theorem for these classes of functions.

The aim of this paper is to define the operator of $q$-derivative based upon the Borel distribution and by using this operator, we familiarize a new subclass of $\beta$-uniformly starlike functions $\beta$-${𝒮}(\alpha ,q,\lambda ,\delta ).$ Further, we obtain coefficient estimates, distortion theorems, convex linear combinations and radii of close-to-convexity, starlikeness and convexity for functions $f\in \beta$-${𝒮}(\alpha ,q,\lambda ,\delta ).$ We also determine the second Hankel inequality for functions belonging to this subclass.

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.

Considering the interesting results obtained recently by studying Rabotnov function, a new investigation is presented in this paper related to the topic of introducing new classes of bi-univalent functions. Using the normalized Rabotnov function and the concept of subordination, a new class of bi-univalent functions ${{\mathscr{H}}}_{\mathrm{\Sigma }}(\alpha ,\beta ,\delta ;\psi )$ is defined and studied regarding coefficient estimates. Bounds of the Taylor–Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ are given and using those estimates, Fekete–Szegö problem is also investigated for the newly introduced class. For particular values of the parameters involved in the definition of the class ${{\mathscr{H}}}_{\mathrm{\Sigma }}(\alpha ,\beta ,\delta ;\psi )$, certain particular classes are obtained and investigated in the corollaries associated to each result proved for the class ${{\mathscr{H}}}_{\mathrm{\Sigma }}(\alpha ,\beta ,\delta ;\psi )$.

In the current work, we alter the Opoola derivative operator using quantum calculus. We examine a novel specific subfamily of analytical functions with complex order connected to the subordination principal by using the modified $q$-Opoola derivative operator. Next, we examine the Fekete–Szegö inequality for the generated subclass in the open unit disk. This paper sheds light the significant connections between the results introduced in this work and previous results.

In this paper, we investigate the growth and regular growth of functions of infinite order defined by Laplace-Stieltjes transformations and obtain some necessary and sufficient conditions.

In this note we study removable singularities for analytic functions in Hardy H^p spaces, in BMO and in locally Lipschitz spaces.