By using a certain general conic domain as well as the quantum (or q-) calculus, here we define and investigate a new subclass of normalized analytic and starlike functions in the open unit disk U . In particular, we find the Hankel determinant and the Toeplitz matrices for this newly-defined class of analytic q-starlike functions. We also highlight some known consequences of our main results.

Hankel and Toeplitz Determinants for a Subclass of q-Starlike Functions Associated with a General Conic Domain

For analytic functions f in the unit disk $${\mathbb {D}}$$ normalized by $$f(0)=0$$ and $$f'(0)=1$$ satisfying in $${\mathbb {D}}$$ respectively the conditions $${{\,\mathrm{Re}\,}}\{ (1-z)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z^2)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z+z^2)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z)^2f'(z) \} > 0,$$ the sharp upper bound of the third logarithmic coefficient in case when $$f''(0)$$ is real was computed.

/pdf/on-the-third-logarithmic-coefficient-in-some-subclasses-of-2eiewztdou.pdf

On the third logarithmic coefficient in some subclasses of close-to-convex functions

Bohr radius for certain classes of starlike and convex univalent functions

Sharp lower and upper bounds for the second- and third-order Hermitian Toeplitz determinants for the classes of starlike and convex functions of order $$\alpha $$ ($$\alpha \in [0,1)$$) are given.

The second and third-order Hermitian Toeplitz determinants for starlike and convex functions of order $$\alpha $$α

In this paper our aim is to study some valuable problems dealing with newly defined subclass of multivalent $ q $-starlike functions. These problems include the initial coefficient estimates, Toeplitz matrices, Hankel determinant, Fekete-Szego problem, upper bounds of the functional $ \left \vert a_{p+1}-\mu a_{p+1}^{2}\right \vert $ for the subclass of multivalent $ q $-starlike functions. As applications we study a $ q $-Bernardi integral operator for a subclass of multivalent $ q $-starlike functions. Furthermore, we also highlight some known consequence of our main results.

Hankel and Toeplitz determinant for a subclass of multivalent $ q $-starlike functions of order $ \alpha $

Univalent Functions: A Primer

Let denote the class of analytic and univalent functions in which are of the form . We determine sharp estimates for the Toeplitz determinants whose elements are the Taylor coefficients of functions in and certain of its subclasses. We also discuss similar problems for typically real functions.

Toeplitz determinants whose elements are the coefficients of analytic and univalent functions

Let S denote the class of analytic univalent functions in 𝔻:= {z ϵ ℂ: |z| < 1} normalized so that $$ f(z)=z+{\sum}_{n=2}^{\infty }{a}_n{z}^n. $$
 Let C and S∗ be the subclasses of S consisting of convex and starlike functions, respectively. For real α, the class M
 α
 of alpha-convex functions f ∈ S defined by $$ \operatorname{Re}\left\{\left(1-a\right)\frac{zf^{\hbox{'}}(z)}{f(z)}+a\left(\frac{zf^{\hbox{'}\hbox{'}}(z)}{f^{\hbox{'}}(z)}+1\right)\right\}>0, $$
 z ϵ 𝔻, is well known, so thatM1 = C andM0 = S
 *
 . We give bounds for the second Hankel determinant $$ {H}_2(2)=\left|{a}_2{a}_4-{a}_3^2\right| $$
 when f ∈ M
 α
 and α ≥ 0, thus extending the well-known results in the cases α = 0 and α = 1. We also give bounds for a wider class of functions.

The second Hankel determinant for alpha-convex functions

Let the function $f$ be analytic in $\\mathbb{D}=\\{z:|z|<1\\}$ and given by $f(z)=z+\\sum _{n=2}^{\\infty }a_{n}z^{n}$ . For $0<\\unicode[STIX]{x1D6FD}\\leq 1$ , denote by ${\\mathcal{C}}(\\unicode[STIX]{x1D6FD})$ the class of strongly convex functions. We give sharp bounds for the initial coefficients of the inverse function of $f\\in {\\mathcal{C}}(\\unicode[STIX]{x1D6FD})$ , showing that these estimates are the same as those for functions in ${\\mathcal{C}}(\\unicode[STIX]{x1D6FD})$ , thus extending a classical result for convex functions. We also give invariance results for the second Hankel determinant $H_{2}=|a_{2}a_{4}-a_{3}^{2}|$ , the first three coefficients of $\\log (f(z)/z)$ and Fekete–Szegö theorems.

Derek K. Thomas

Papers

Univalent Functions: A Primer

Toeplitz determinants whose elements are the coefficients of analytic and univalent functions

The second Hankel determinant for alpha-convex functions

Invariance of the coefficients of strongly convex functions

Hankel determinants for starlike and convex functions associated with sigmoid functions