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Abstract
In this paper we consider some properties of Jackson's difference operator for convex univalent functions in $|z|<1$ with complex parameter $q$ as a Hadamard product of two power series. Jackson in 1908 introduced for a real $q$, $q\in[0,1)$, the difference operator \mbox{${\rm d}_qf(z)$} for an analytic function $f$ in the unit disc $|z|<1$ in the complex plane. Thanks to this operator, many mathematicians have extended the theory of functions in $q$-theory. The $q$-theory has found many applic
Results & Lemmas (7)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Theorem 3.1
Theorem 3.1. If f is in the class K of convex univalent functions, then (3.1) for all and,. The result is the best possible.
Theorem 3.1. If f is in the class K of convex univalent functions, then
(3.1)
$$\Re \left\{ \frac{\zeta}{1-\zeta} \left( \frac{f'(z)}{\zeta d_{\zeta} f(z)} - 1 \right) \right\} > \frac{1}{2}$$
for all $z \in \mathbb{D}$ and $\zeta$ , $|\zeta| \leq 1$ . The result is the best possible.
Corollary 3.2
Corollary 3.2. If f is in the class K of convex univalent functions, then (3.10) and (3.11) for all z ∈ D and ζ such that |ζ| ≤ 1. The…
Corollary 3.2. If f is in the class K of convex univalent functions, then
(3.10)
$$\Re \left\{ \frac{f'(z)}{(1-\zeta)\mathrm{d}_{\zeta}f(z)} \right\} > \Re \left\{ \frac{1+\zeta}{2(1-\zeta)} \right\} > 0$$
and
(3.11)
$$\Re \left\{ \frac{1}{1-\zeta} \left( \frac{f'(z)}{\mathrm{d}_{\zeta} f(z)} - 1 \right) \right\} > -\frac{1}{2}$$
for all z ∈ D and ζ such that |ζ| ≤ 1. The results are the best possible.
In [\[6,](#page-7-7) Lemma 2] it was proved that for f ∈ K
(3.12)
$$\Re \left\{ \frac{f'(z)}{(1-\zeta)\mathrm{d}_{\zeta}f(z)} \right\} > 0$$
for all z, ζ ∈ D. Therefore, [\(3.10\)](#page-4-0) improves earlier result [\(3.12\)](#page-4-1).
Corollary 3.3
Corollary 3.3. If f is in the class K of convex univalent functions, then (3.13) The result is the best possible.
Corollary 3.3. If f is in the class K of convex univalent functions, then
(3.13)
$$\Re \left\{ \frac{f'(z)}{(1-\zeta)\mathrm{d}_{\zeta}f(z)} \right\} > \frac{1-|\zeta|^2}{2(1-2\Re \epsilon \zeta + |\zeta|^2)} = \frac{1-|\zeta|^2}{2|1-\zeta|^2}.$$
The result is the best possible.
Corollary 3.4
Corollary 3.4. If f is in the class K of convex univalent functions and, then (3.14) for all, Equivalently, implies. The result is the best…
Corollary 3.4. If f is in the class K of convex univalent functions and $q \in [0,1)$ , then
(3.14)
$$\Re\left\{\frac{f'(z)}{\mathrm{d}_q f(z)}\right\} > \frac{1+q}{2}$$
for all $z \in \mathbb{D}$ , Equivalently, $f \in \mathcal{K}$ implies $f \in \mathcal{R}(q, (1+q)/2)$ . The result is the best possible.
Open problem. Is it true that if f is in the class $\mathcal{K}$ of convex univalent functions, then
(3.15)
$$\Re \left\{ \frac{f'(z)}{\mathrm{d}_{\zeta}f(z)} \right\} > \frac{1+|\zeta|}{2} ?$$
For q = 0 inequality (3.14) becomes
$$\Re \left\{ \frac{zf'(z)}{f(z)} \right\} > \frac{1}{2}$$
for all $z \in \mathbb{D}$ , this is a well known result for convex univalent function f.
Corollary 3.5
Corollary 3.5. If f is in the class K of convex univalent functions and, then there exists an analytic function, p(0) = 1, such that (3.16)…
Corollary 3.5. If f is in the class K of convex univalent functions and $q \in [0,1)$ , then there exists an analytic function $p \in \mathcal{H}$ , p(0) = 1, such that
(3.16)
$$f'(z) = \left(\frac{1-q}{2}p(z) + \frac{1+q}{2}\right) d_q f(z), \quad z \in \mathbb{D}$$
and
$$\mathfrak{Re}\left\{p(z)\right\}>0,\quad z\in\mathbb{D}.$$
Theorem 3.6
Theorem 3.6. If f is in the class K of convex univalent functions, then and (3.18) for all z, q such that. The result is sharp.
Theorem 3.6. If f is in the class K of convex univalent functions, then
$$(3.17) \qquad \frac{1+qr}{1+r} \le \Re\left\{\frac{f'(z)}{\mathrm{d}_{\sigma}f(z)}\right\} \le \frac{1-qr}{1-r}$$
and
(3.18)
$$\frac{1+qr}{1+r} \le \left| \frac{f'(z)}{\mathrm{d}_q f(z)} \right| \le \frac{1-qr}{1-r}$$
for all z, q such that $|z| = r < 1, q \in [0, 1)$ . The result is sharp.
Theorem 3.7
Theorem 3.7. If f is in the class K of convex univalent functions, then for all z, q such that,.
Theorem 3.7. If f is in the class K of convex univalent functions, then
$$\Re \left\{ \frac{f(z)}{\left[\log \frac{1}{1-z}\right] * z \operatorname{d}_q f(z)} \right\} > \frac{1+q}{2}$$
for all z, q such that $z \in \mathbb{D}$ , $q \in [0, 1)$ .
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