Weighted visual secret sharing with multiple decryptions and lossless recovery

1.   Introduction

Let $A$ denote the class of functions of the form

which are analytic in the open unit disk $D = \left(z:\mid z\mid < 1\right)$ and normalized by $f(0) = 0$ and $f^{\prime}(0) = 1$. Recall that, $S\subset A$ is the univalent function in $D = \left(z:\mid z\mid < 1\right)$ and has the star-like and convex functions as its sub-classes which their geometric condition satisfies $Re\left(\dfrac{zf^{\prime}(z)}{f(z)}\right) > 0$ and $Re\left(1+ \dfrac{zf^{\prime \prime}(z)}{f^{\prime}(z)}\right) > 0$. The two well-known sub-classes have been used to define different subclass of analytical functions in different direction with different perspective and their results are too voluminous in literature.

Two functions $f$ and $g$ are said to be subordinate to each other, written as $f\prec g$, if there exists a Schwartz function $w(z)$ such that

where $w(0)$ and $\mid w(z) \mid < 1$ for $z\; \epsilon \; D$. Let $P$ denote the class of analytic functions such that $p(0) = 1$ and $p(z)\prec \dfrac{1+z}{1-z}$, $z\; \epsilon \; D$. See ^[1] for details.

Goodman ^[2] proposed the concept of conic domain to generalize convex function which generated the first parabolic region as an image domain of analytic function. The same author studied and introduced the class of uniformly convex functions which satisfy

In recent time, Ma and Minda ^[3] studied the underneath characterization

The characterization studied by ^[3] gave birth to first parabolic region of the form

which was later generalized by Kanas and Wisniowska (^[5,6]) to

The $\Omega_{k}$ represents the right half plane for $k = 0$, hyperbolic region for $0 < k < 1$, parabolic region for $k = 1$ and elliptic region for $k > 1$ ^[30].

The generalized conic region (1.5) has been studied by many researchers and their interesting results litter everywhere. Just to mention but a few Malik ^[7] and Malik et al. ^[8].

More so, the conic domain $\Omega$ was generalized to domain $\Omega[A, B]$, $-1\leq B < A\leq1$ by Noor and Malik ^[9] to

and it is called petal type region.

A function $p(z)$ is said to be in the class $UP[A, B]$, if and only if

where $\tilde{p}(z) = 1 + \dfrac{2}{\pi^2} \left(log \dfrac{1+\sqrt{z}}{1-\sqrt{z}} \right)^2$.

Taking $A = 1$ and $B = -1$ in (1.8), the usual classes of functions studied by Goodman ^[1] and Kanas (^[5,6]) will be obtained.

Furthermore, the classes $UCV[A, B]$ and $ST[A, B]$ are uniformly Janoski convex and Starlike functions satisfies

and

or equivalently

and

Setting $A = 1$ and $B = -1$ in (1.7) and (1.8), we obtained the classes of functions investigated by Goodman ^[2] and Ronning ^[10].

The relevant connection to Fekete-Szegö problem is a way of maximizing the non-linear functional $\left| a_3 - \lambda a_2^2\right|$ for various subclasses of univalent function theory. To know much of history, we refer the reader to ^{[11,12,13,14]} and so on.

The error function was defined because of the normal curve, and shows up anywhere the normal curve appears. Error function occurs in diffusion which is a part of transport phenomena. It is also useful in biology, mass flow, chemistry, physics and thermomechanics. According to the information at hand, Abramowitz ^[15] expanded the error function into Maclaurin series of the form

The properties and inequalities of error function were studied by ^[16] and ^[4] while the zeros of complementary error function of the form

was investigated by ^[17], see for more details in ^[18,19] and so on. In recent time, ^[20,21,22] and ^[23] applied error functions in numerical analysis and their results are flying in the air.

For $f$ given by ^[15] and $g$ with the form $g(z) = z+b_{2}z^{2}+b_{3}z^{3}+ \cdots$ their Hadamard product (convolution) by $f*g$ and at is defined as:

Let $E_{r}f$ be a normalized analytical function which is obtained from (1.9) and given by

Therefore, applying a notation (1.11) to (1.1) and (1.12) we obtain

where $E_rf$ is the class that consists of a single function or $E_rf$. See concept in Kanas et al. ^[18] and Ramachandran et al. ^[19].

Babalola ^[24] introduced and studied the class of $\lambda -$pseudo starlike function of order $\beta (0 \leq \beta \leq 1)$ which satisfy the condition

where $\lambda \geq 1 \in \Re (z \in D)$ and denoted by $\angle_\lambda(\beta)$. We observed from (1.14) that putting $\lambda = 2$, the geometric condition gives the product combination of bounded turning point and starlike function which satisfy

Olatunji ^[25] extended the class $\angle_\lambda(\beta)$ to $\angle_\lambda^\beta (s, t, \varPhi)$ which the geometric condition satisfy

where $s, t \in C, s \neq t, \lambda \geq 1 \in \Re, 0 \leq \beta < 1, z \in D$ and $\varPhi (z)$ is the modified sigmoid function. The initial coefficient bounds were obtained and the relevant connection to Fekete-Szegö inequalities were generated. The contributions of authors like Altinkaya and Özkan ^[26] and Murugusundaramoorthy and Janani ^[27] and Murugusundaramoorthy et al. ^[28] can not be ignored when we are talking on $\lambda$-pseudo starlike functions.

Inspired by earlier work by ^[18,19,29]. In this work, the authors employed the approach of ^[13] to study the coefficient inequalities for pseudo certain subclasses of analytical functions related to petal type region defined by error function. The first few coefficient bounds and the relevant connection to Fekete-Szegö inequalities were obtained for the classes of functions defined. Also note that, the results obtained here has not been in literature and varying of parameters involved will give birth to corollaries.

For the purpose of the main results, the following lemmas and definitions are very necessary.

Lemma 1.1. If $p(z) = 1 + p_1z + p_2 z^2 + \cdots$ is a function with positive real part in $D$, then, for any complex $\mu$,

and the result is sharp for the functions

Lemma 1.2. ^[29] Let $p \in UP[A, B], -1 \leq B < A \leq 1$ and of the form $p(z) = 1 + {\sum\limits_{n = 1}^{\infty}}p_nz^n$. Then, for a complex number $\mu$, we have

The result is sharp and the equality in (1.15) holds for the functions

or

Proof. For $h \in P$ and of the form $h(z) = 1 + {\sum\limits_{n = 1}^{\infty}}c_nz^n$, we consider

where $w(z)$ is such that $w(0) = 0$ and $|w(z)| < 1$. It follows easily that

Now, if ${\rm{\tilde p(z)}} = 1+R_{1}z+R_{2}z^{2}+\cdots$, then from (1.16), one may have,

where $R_{1} = \frac{8}{ \pi^{2}}, \; R_{2} = \frac{16}{3 \pi^{2}},$ and $R_{3} = \frac{184}{ 45\pi^{2}}$, see ^[30]. Substitute $R_{1}, R_{2}$ and $R_{3}$ into (1.17) to obtain

Since $p\in UP[A, B]$, so from relations (1.16), (1.17) and (1.18), one may have,

This implies that,

If $p(z) = 1+\sum_{n = 1}^{\infty}p_{n}z^{n}$, then equating coefficients of $z$ and $z^{2}$, one may have,

and

Now for a complex number $\mu$, consider

This implies that

Using Lemma 1.1, one may have

where $v = \frac{1}{6}+\frac{2(B+1)}{\pi^{2}}+\frac{2\mu(A-B)}{\pi^{2}}$, which completes the proof of the Lemma.

Definition 1.3. A function $\mathcal{F}\epsilon A$ is said to be in the class $UCV[\lambda, A, B]$, $-1\leq B < A\leq1$, if and only if,

where $\lambda \geq 1\epsilon R$ or equivalently $\frac{(z(\mathcal{F}'(z)^{\lambda}))^{\prime}}{\mathcal{F}'(z)}\epsilon UP[A, B]$.

Definition 1.4. A function $\mathcal{F}\epsilon A$ is said to be in the class $US[\lambda, A, B]$, $-1\leq B < A\leq1$, if and only if,

where $\lambda \geq 1\epsilon R$ or equivalently $\frac{z(\mathcal{F}'(z)^{\lambda})}{\mathcal{F}(z)}\epsilon UP[A, B]$.

Definition 1.5. A function $\mathcal{F}\epsilon A$ is said to be in the class $UM_{\alpha}[\lambda, A, B]$, $-1\leq B < A\leq1$, if and only if,

where $\alpha \geq 0$ and $\lambda \geq 1\epsilon R$ or equivalently $(1-\alpha)\frac{z(\mathcal{F}'(z)^{\lambda})}{f(z)}+\alpha\frac{(z(f'(z)^{\lambda}))^{\prime}}{f'(z)}\in UP[A, B]$.

2.   Main results

In this section, we shall state and prove the main results, and several corollaries can easily be deduced under various conditions.

Theorem 2.1. Let $\mathcal{F}\in US[\lambda, A, B]$, $-1\leq B < A\leq1$, and of the form (1.13). Then, for a real number $\mu$, we have

Proof. If $\mathcal{F}\in US[\lambda, A, B]$, $-1\leq B < A\leq1$, the it follows from relations (1.18), (1.19), and (1.20),

where $w(z)$ is such that $w (0) = 0$ and $\mid w(z) \mid < 1$. The right hand side of the above expression get its series form from (1.13) and reduces to

If $\mathcal{F}(z) = z + \sum_{n = 2}^{\infty}\frac{(-1)^{n-1}a_{n}z^{n}}{(2n-1)(n-1)!}$, then one may have

From (2.1) and (2.2), comparison of coefficient of $z$ and $z^{2}$ gives,

and

This implies, by using (2.3), that

Now, for a real number $\mu$ consider

where $v = \frac{1}{6}+\frac{2(B+1)}{\pi^{2}}-\frac{(A-B)}{(1-2\lambda)^{2}\pi^{2}}\left(2(2\lambda^{2}-4\lambda+1) -\frac{9\mu(1-3\lambda)}{5}\right)$.

Theorem 2.2. Let $\mathcal{F}\in UCV[\lambda, A, B]$, $-1\leq B < A\leq1$, and of the form (1.13). Then, for a real number $\mu$, we have

Proof. If $\mathcal{F}\in UCV[\lambda, A, B]$, $-1\leq B < A\leq1$, then it follows from relations (1.18), (1.19), and (1.21),

where $w(z)$ is such that $w (0) = 0$ and $\mid w(z) \mid < 1$. The right hand side of the above expression get its series form from (1.13) and reduces to,

If $\mathcal{F}(z) = z+\sum \dfrac{(-1)^{n-1}a_{n}z^{n}}{(2n-1)(n-1)!},$ then we have,

From (2.4) and (2.5), comparison of coefficients of $z$ and $z^{2}$ gives,

and

This implies, by using (2.6), that

Now, for a real number $\mu$, consider

where

Theorem 2.3. $\mathcal{F}\in M_{\alpha}[\lambda, A, B]$, $-1\leq B < A\leq1$, $\alpha\geq 0$ and of the form (1.13). Then, for a real number $\mu$, we have

Proof. Let $\mathcal{F}\in M_{\alpha}[\lambda, A, B]$, $-1\leq B < A\leq1$, $\alpha\geq 0$ and of the form (1.13). Then, for a real number $\mu$, we have

where $w(z)$ is such that $w(z_0) = 0$ and $|w(z)| < 1$. The right hand side of the above expression get its series form from (2.7) and reduces to

If $\mathcal{F}(z) = z+ \sum_{n = 2}^{\infty} \dfrac{(-1)^{n-1} a_{n}z^{n}}{(2n-1)(n-1)!}$, then one may have

from (2.8) and (2.9), comparison of coefficients of $z$ and $z^{2}$ gives

and

This implies, by using (2.10), that

Now, for a real number $\mu$, consider

where

3.   Conclusions

The force applied on certain subclasses of analytical functions associated with petal type domain defined by error function has played a vital role in this work. The results obtained are new and varying the parameters involved in the classes of function defined, these will bring new more results that has not been in existence.

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions.

Conflict of interest

The authors declare that they have no conflict of interests.