On a generalization of starlike functions

Theorem 2.1. Let a, b, c be non-negative real numbers with 0 < 1 − aq < 1 − cq and 0 < 1 − b < 1 − c. For 0 < q < 1 and r ∈ (0, 1], the function z 7→ zΦ[a, b; c; q, rz] has the order of q-starlikeness $$\sigma_q(z\Phi[a,b;c;q,rz]) = 1 + \rho q \frac{(1-a)(1-b)}{(1-c)(1-q)} \frac{\Phi[aq,bq;cq;q,\rho]}{\Phi[a,b;c;q,\rho]}$$ where $$\rho = -r \text{ if } \frac{q(1-a)}{a(1-q)} =: s > 0 \text{ and } \rho = r \text{ if } s < 0.$$ In particular, we have $$1 + \frac{s\rho}{1 - \rho} \le \sigma_q(z\Phi[a, b; c; q, rz]) \le 1 + \frac{\rho s(1 - b)}{2(1 - c)}.$$ From this we can observe the following Remark 2.2. The case s < 0 with r = 1 in Theorem [2.1](#page-2-1) is considered in the limiting sense. In this case, the lower bound 1 + sr 1 − r is equal to −∞. Proof of Theorem 2.1. Set $\Phi(z) = \Phi[a, b; c; q, z]$ and $f(z) = z\Phi(z)$ . Now, by (1.3) we have $$\begin{array}{lcl} (D_q f)(z) & = & \dfrac{\Phi(z) - q \Phi(qz)}{1 - q} \\ \\ & = & \dfrac{\Phi(z) - q \Phi(z) + q \Phi(z) - q \Phi(qz)}{1 - q} \\ \\ & = & \dfrac{\Phi(z)(1 - q) + q(\Phi(z) - \Phi(qz))}{1 - q} \\ \\ & = & \Phi(z) + zq(D_q \Phi)(z). \end{array}$$ Hence. (2.1) $$w = \frac{z(D_q f)(z)}{f(z)} = 1 + zq \frac{(1-a)(1-b)}{(1-c)(1-q)} \frac{\Phi[aq, bq; cq; q, z]}{\Phi[a, b; c; q, z]},$$ where the last equality holds by [7, 1.12(ii), pp. 27]. Recall the difference equation stated in [1], which is equivalent to $$\frac{\Phi[aq,bq;cq;q,z]}{\Phi[a,b;c;q,z]} = \frac{(1-c)}{a(1-b)z} \left[ \frac{\Phi[aq,b;c;q,z]}{\Phi[a,b;c;q,z]} - 1 \right].$$ Substituting this ratio in (2.1), we get $$w = 1 + s \left[ \frac{\Phi[aq, b; c; q, z]}{\Phi[a, b; c; q, z]} - 1 \right] = 1 - s + s \frac{\Phi[aq, b; c; q, z]}{\Phi[a, b; c; q, z]},$$ where s is defined in the statement of our theorem with $q \in (0,1)$ . It follows from [1] that w has an integral representation (2.2) $$w = 1 - s + s \int_0^1 \frac{1}{1 - tz} d\mu(t),$$ with the non-negative real numbers a,b,c satisfying the conditions $0 \le 1 - aq \le 1 - cq$ and 0 < 1 - b < 1 - c. Now, for s > 0, $r \in (0,1]$ and from equation (2.2) it follows that the minimum of Re w for $|z| \le r$ is attained at the point z = -r and that the minimum is $1 - \frac{rs}{(1+r)}$ . Secondly, for s < 0, $r \in (0,1]$ and from equation (2.2), it follows that the minimum of Re w for $|z| \le r$ is attained at the point z = r and that the minimum is $1 + \frac{rs}{(1-r)}$ . This in combination with (2.1), yields the order of q-starlikeness of $z\Phi[a,b;c;q,rz]$ . The upper estimate for Re w follows from (2.1) and an integral representation of the ratio $\Phi[aq,bq;cq;q,z]/\Phi[a,b;c;q,z]$ obtained in [1, Theorem 2.13]. Hence, the conclusion of our theorem follows immediately. Remark 2.3. Making the substitutions $a \to q^a$ , $b \to q^b$ and $c \to q^c$ , and taking the limit as $q \to 1^-$ , we achieve the result of Küstner [15, Theorem 1.1] as special case.

Corollary 2.4. Let $q \in (0,1)$ , a,b,c be non-negative real numbers such that $1 > a \ge b \ge c$ and $r \in (0,1]$ . Then $$z\Phi[a,b;c;q,rz] \in \mathcal{S}_q\left(1-\frac{rs}{1+r}\right),$$ where $$s = \frac{q(1-a)}{a(1-q)}$$ .

Corollary 2.5. Let a, b, c be non-negative real numbers such that $1 > a \ge b \ge c$ . Then $$z\Phi[a,b;c;q,z] \in \mathcal{S}_q(1-s/2),$$ where $$s = \frac{q(1-a)}{a(1-q)}$$ . In particular, when s=2, the shifted basic hypergeometric function $z\Phi[a,b;c;q,z] \in \mathcal{S}_q$ . When we put $a=q^a,b=q^b$ and $c=q^c$ and allow $q\to 1^-$ , then Corollary 2.5 leads to an well-known result on the order of starlikeness of the Gaussian hypergeometric functions namely,

Corollary 2.6. [25, Theorem B] Let a, b, c be non-negative real numbers such that $a \le b \le c$ . Then $$zF(a, b; c; z) \in \mathcal{S}^*(1 - a/2).$$ This result on Gaussian hypergeometric function of Corollary 2.6 is not only interesting by itself, but also useful and employed for further research in geometric function theory. Many researchers used this result, particularly Ponnusamy and Sahoo in [23] used it innovatively to study pre-Schwarzian norm estimates for integral operators (defined by convolution) of functions belonging to special subclasses of the class of univalent functions with hypergeometric functions. Similarly, we expect that these results on basic hypergeometric functions will be fruitful and pave the way for further research in function theory as well as in physics.

Lemma 3.1. [16, pp. 254-256] Let the function $p \in \mathcal{P}$ and be given by the power series (3.1). Then $$2p_2 = p_1^2 + x(4 - p_1^2),$$ $$4p_3 = p_1^3 + 2(4 - p_1^2)p_1x - p_1(4 - p_1^2)x^2 + 2(4 - p_1^2)(1 - |x|^2)z,$$ for some x and z satisfying $|x| \le 1$ , $|z| \le 1$ , and $p_1 \in [0,2]$ .

Lemma 3.2. [19, Lemma 1] Let the function $p \in \mathcal{P}$ and be given by the power series (3.1). Then for any real number $\lambda$ , $$|p_2 - \lambda p_1^2| \le 2 \max\{1, |2\lambda - 1|\}$$ and the result is sharp.

Lemma 3.3. (Carathédory lemma). If a function $p(z) = 1 + \sum_{n=1}^{\infty} p_n z^n \in \mathcal{P}$ , then $|p_n| \leq 2, n = 1, 2, \ldots$ The result is sharp for $$p(z) = \frac{1+z}{1-z} = 1 + \sum_{n=1}^{\infty} 2z^n.$$ 3.1. The Bieberbach-type problem. The Bieberbach-type problem for the class $S_q$ is investigated and we prove the following result.

Theorem 3.4. If $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_q$ , then for all $n \geq 2$ , $$|a_n| \le \prod_{j=2}^n \left( \frac{\frac{1-q^{j-1}}{1-q}+1}{\frac{1-q^j}{1-q}-1} \right).$$ Equality holds for the function F satisfying $z(D_qF)(z)/F(z) = (1+z)/(1-z)$ .

Theorem 3.5. Let $f \in \mathcal{S}_q$ be of the form (1.1) and $\mu$ be any complex number. Then $$|a_3 - \mu a_2^2| \le \max \left\{ \left| \frac{2(2+q) - 4\mu(1+q)}{q^2(1+q)} \right|, \frac{2}{q(1+q)} \right\}.$$ Equality occurs for the functions F and G satisfying $$\frac{z(D_q F)(z)}{F(z)} = \frac{1+z}{1-z}$$ and (3.8) $$\frac{z(D_qG)(z)}{G(z)} = \frac{1+z^2}{1-z^2}.$$

Theorem 3.7. Let $f \in S_q$ be of the form (1.1). Then $$|H_2(2)| = |a_2a_4 - a_3^2| \le \frac{4}{q^2(1+q)^2}.$$ Equality occurs for the function G(z) defined in (3.8).

Definition 1.1. A function $f \in \mathcal{A}$ is said to be in the class of q-starlike functions of order $\alpha$ , denoted by $\mathcal{S}_q(\alpha)$ , $0 \le \alpha < 1$ , if Re $$\left(\frac{z(D_q f)(z)}{f(z)}\right) > \alpha, \quad z \in \mathbb{D}.$$ Clearly, $S_q(\alpha) \supset S_q^(\alpha)$ and as $q \to 1^-$ , $S_q(\alpha) = S^(\alpha)$ . Set $S_q := S_q(0)$ so that as $q \to 1^-$ , $S_q(0) = S^*$ . In Section 2, we establish a result on the order of q-starlikeness of shifted basic hypergeometric functions $$z\Phi[a,b;c;q,z] = z \sum_{n=0}^{\infty} \frac{(a;q)_n(b;q)_n}{(c;q)_n(q;q)_n} z^n, \quad z \in \mathbb{D},$$ where $(a;q)_n = (1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1})$ , $(a;q)_0 = 1$ with $0 \le q < 1$ , a,b,c are real parameters and $(c;q)_n \ne 0$ . For the basic properties of Heine's hypergeometric functions the basic references are [3,27]. Interestingly, the replacements of a,b and c by $q^a,q^b$ and $q^c$ respectively, then as $q \to 1^-$ , the function $\Phi[q^a,q^b;q^c;q,z]$ tends to the well-known Gaussian hypergeometric functions $$zF(a,b;c;z) = z \sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_n(1)_n} z^n, \quad z \in \mathbb{D},$$ where a, b, c are real parameters, (a)⁰ = 1, (a)ⁿ = a(a+ 1)· · ·(a+n−1) is the Pochhammer symbol and c is neither 0 nor a negative integer (except in special cases where a = −m or b = −m and c = −p with p > m).