Convolution theorems for starlike and convex functions in the unit disc

Theorem 1.1. We have the following inclusions: (i) $$M_{1,\lambda}^{(1)} \subset S^*$$ for $\lambda = 1/2(\log 4 - 1)$ , (ii) $$M_{1,\lambda}^{(1)} \subset \mathcal{K}$$ for $\lambda = 1/\log 4$ , (iii) $$M_{1,\lambda}^{(2)} \subset S^*$$ for $\lambda = 1/[2 - 4 \log 2 + \pi^2/6]$ , (iv) $$M_{1,\lambda}^{(2)} \subset \mathcal{K}$$ for $\lambda = 1/(\log 4)^2$ , (v) $$M_{1,\lambda} M_{1,\lambda} \subset S$$ for $2\lambda^2 = 1/[2 - 4\log 2 + \pi^2/6]$ , (vi) $$M_{1,\lambda} * M_{1,\lambda} \subset \mathcal{K}$$ for $2\lambda^2 = 1/(\log 4)^2$ , (vii) $$M_{1,\lambda} \mathcal{P}_0^0 \subset \mathcal{S}$$ for $2\lambda = 1/2(\log 4 - 1)$ , (viii) $$M_{1,\lambda} * \mathcal{P}_0^0 \subset \mathcal{K}$$ for $2\lambda = 1/2 \log 4$ . Cases (i) and (ii) of Theorem 1.1 are Theorem 2 in [8], whereas (iii) and (iv) are Theorem 1 in [8]. Similarly (v) and (vi) are Theorem 3 in [8] and (vii) and (viii) are Theorem 4 in [8]. The main aim of this paper is to obtain more general theorems which, in particular, yield all the results of Theorem 1.1 as special cases. We also obtain some additional results concerning the class $M_{1,\lambda}^{(3)}$ . To prove these results we shall use a number of lemmas that have been obtained recently with the help of duality theory for convolutions. For results in duality theory, we refer to the book by Ruscheweyh [6]. LEMMA 1.2 ([2, Theorem 2.1]). Let $\alpha : [0,1] \to \mathbb{R}$ be nonnegative, integrable with $\int_0^1 \alpha(t) dt = 1$ and suppose that $\Lambda(t) = \int_t^1 \alpha(s) s^{-2} ds$ satisfies the following conditions: - (i) $\Lambda$ is not integrable on [0,1], $t\Lambda(t)$ is integrable on [0,1], and positive on (0,1). - (ii) For $0 \le \gamma \le 1/2$ , $$\frac{\Lambda(t)}{(1+t)(1-t)^{1+2\gamma}}$$ is decreasing on (0,1). For $\beta > 0$ , define (1.3) $$F(z) = z \int_{0}^{1} \left(1 + \frac{z\beta}{1 - tz}\right) \alpha(t) dt * \phi(z),$$ where $\phi \in \mathcal{P}_0$ . Then $F \in \mathcal{S}_{\gamma}$ for all $\gamma \in [0, 1/2]$ and for $\beta$ given by (1.4) $$\beta \int_{0}^{1} \left( 1 - \frac{1}{(1 - \gamma)(1 + t)} + \frac{\gamma}{1 - \gamma} \frac{\log(1 + t)}{t} \right) \frac{\alpha(t)}{t} dt = \frac{1}{2}.$$ A combination of Theorem 2.13 and Lemma 1.4 of [2] gives the following result.

Lemma 1.5. Let $\alpha:[0,1]\to\mathbb{R}$ be nonnegative with $\int_0^1 \alpha(t) dt=1$ and suppose that $\Lambda(t)=\alpha(t)/t$ satisfies the following conditions: - (i) $\Lambda$ is not integrable on [0,1], $t\Lambda(t)$ is integrable on [0,1], and positive on (0,1). - (ii) For $0 \le \gamma \le 1/2$ , $$\frac{\Lambda(t)}{(1+t)(1-t)^{1+2\gamma}}$$ is decreasing on (0,1). For $\beta > 0$ , let F be defined by (1.3). Then $F \in \mathcal{K}_{\gamma}$ for all $\gamma \in [0, 1/2]$ and for $\beta$ given by $$\beta \int_{0}^{1} \left( 1 - \frac{1 - \gamma(1+t)}{(1-\gamma)(1+t)^{2}} \right) \frac{\alpha(t)}{t} dt = \frac{1}{2}.$$ Earlier results which led to results such as Lemmas 1.2 and 1.5 can be found in Fournier and Ruscheweyh [4] and Ponnusamy and Rønning [5]. LEMMA 1.6 ([2, Theorem 2.10]). If $\beta > 0$ and $F \in M_{1,\beta}$ then $F \in \mathcal{S}_{\gamma}$ $(0 \leq \gamma \leq 1/2)$ whenever $$0 \le \beta \le \frac{1 - 2\gamma}{2\gamma + \log 4}.$$ The case $\gamma = 0$ is due to Ali, Ponnusamy and Singh [1]. LEMMA 1.7 ([7]). If $f, g \in \mathcal{H}$ and $F, G \in \mathcal{K}$ are such that $f \prec F$ and $g \prec G$ , then $f g \prec F G$ . Here $\mathcal{K}$ denotes the family of convex functions (not necessarily normalized) in $\Delta$ .

Theorem 2.1. Let $\gamma \in [0, 1/2]$ and let $\lambda > 0$ be given by (2.2) $$\lambda = \frac{1 - \gamma}{2 - 2\log 4 + \pi^2/6 + (4\log 4 - 6)\gamma}.$$ Then $M_{1,\lambda}^{(2)} \subset \mathcal{S}_{\gamma}$ .

Theorem 2.7. Let $\gamma \in [0, 1/2]$ and let $\lambda > 0$ be given by (2.8) $$\lambda = \frac{1 - \gamma}{2(\log 4 - 1) + \gamma(4 - 4\log 2 - \pi^2/6)}.$$ Then $M_{1,\lambda}^{(1)} \subset \mathcal{S}_{\gamma}$ .

Theorem 2.10. Let $\gamma \in [0, 1/2]$ and let $\lambda > 0$ be given by (2.11) $$\lambda = \frac{1 - \gamma}{2(4\log 2 - 1 - \pi^2/6) - \gamma(10 + 3\zeta(3) - 16\log 2 - \pi^2/3)},$$ where $\zeta(3) = 1.202056903...$ denotes the Riemann zeta value at 3. Then $M_{1,\lambda}^{(3)} \subset \mathcal{S}_{\gamma}$ .

Theorem 3.1. Let $\gamma \in [0, 1/2]$ and let $\lambda > 0$ be given by (3.2) $$\lambda = \frac{1 - 2\gamma}{(\log 4)^2 + 2\gamma \log 4}.$$ Then $M_{1,\lambda}^{(2)} \subset \mathcal{K}_{\gamma}$ .

Theorem 3.4. Let $\gamma \in [0, 1/2]$ and let $\lambda > 0$ be given by $$\lambda = \frac{1 - 2\gamma}{\log 4 + 2\gamma}.$$ Then $M_{1,\lambda}^{(1)} \subset \mathcal{K}_{\gamma}$ .

Theorem 3.5. Let $\gamma \in [0, 1/2]$ and let $\lambda > 0$ be given by (3.6) $$\lambda = \frac{1 - \gamma}{2(3 - \log 16) + \gamma(2 - 2\log 16 + \pi^2/3)}.$$ Then $M_{1,\lambda}^{(3)} \subset \mathcal{K}_{\gamma}$ .

Theorem 4.7. Let $\gamma \in [0, 1/2]$ and $\beta < 1$ . Then: (i) $M_{1,\alpha} * \mathcal{P}^0_{\beta} \subset \mathcal{S}_{\gamma}$ whenever $$2\alpha(1-\beta) = \frac{1-\gamma}{2(\log 4 - 1) + \gamma(4 - 4\log 2 - \pi^2/6)}.$$ (ii) $M_{1,\alpha} * \mathcal{P}^0_{\beta} \subset \mathcal{K}_{\gamma}$ whenever $$2\alpha(1-\beta) = \frac{1-2\gamma}{2\gamma + \log 4}.$$

Theorem 4.8. Let $\gamma \in [0, 1/2]$ and $\alpha, \beta > 0$ . Then: (i) $M_{1,\alpha} * M_{1,\beta} \subset \mathcal{S}_{\gamma}$ whenever $$2\alpha\beta = \frac{1 - \gamma}{2 - 2\log 4 + \pi^2/6 + (4\log 4 - 6)\gamma}.$$ (ii) $M_{1,\alpha} * M_{1,\beta} \subset \mathcal{K}_{\gamma}$ whenever $$2\alpha\beta = \frac{1 - 2\gamma}{(\log 4)^2 + 2\gamma \log 4}.$$