Narrow Results

In this paper, we define two families ${{𝒯}}_{\mathrm{\Sigma }}(\lambda ;r)$ and ${{ℱ}}_{\mathrm{\Sigma }}(\delta ,r)$ of bi-Bazilevič and bi-Ozaki-close-to-convex functions associated with Lucas-balancing polynomials. We demonstrate the upper bounds for the initial Taylor–Maclaurin coefficients. In addition, the Fekete–Szegö type inequalities are derived for functions in these families. Moreover, we indicate certain special cases and consequences for our results.

In this paper, we solve Fekete–Szegö problem for a Ma–Minda type class $M_{{}_{\mathrm{\Sigma }}}^{{}_{a,b,c}}(\lambda ,\nu ,{\psi }_1,{\psi }_2)$ of bi-univalent functions associated with the Hohlov operator which is a specialized case of the widely-investigated Dziok–Srivastava operator which, in turn, is contained in the Srivastava–Wright operator. Results for some of its consequent classes ${{𝒮}{𝒯}}_{\mathrm{\Sigma }}({\psi }_1,{\psi }_2),$${{𝒦}}_{\mathrm{\Sigma }}({\psi }_1,{\psi }_2),$${{ℳ}}_{\mathrm{\Sigma }}({\psi }_1,{\psi }_2)$ and $M_{{}_{\mathrm{\Sigma }}}^{{}_{a,b,c}}(\lambda ,{\psi }_1)$ are also given. A function $f$ in this class may satisfy one subordinate condition, whereas $f^{-1}$, another. One result for that is also given by considering certain ${\psi }_1,{\psi }_2$. Some useful consequences and relationships with some of the known results are also pointed out.

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.

New subclasses of analytic functions are defined by using the function $f_{\delta }(z)$ given by the integral transform $f_{\delta }(z)={\int }_0^z{(\frac{1+r}{1-r})}^{\delta }\frac{1}{1-r^2}dr$. Bounds for the Fekete–Szegö coefficient functional $|a_3-\mu a_2^2|$ for these classes of functions are derived.

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.

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.