Strong differential subordination properties for analytic functions involving the Komatu integral operator

and let ${\mathcal{H}}_0\equiv \mathcal{H}[0,1]$. We also denote by $\mathcal{A}$ the subclass of $\mathcal{H}[a,1]$ with the usual normalization $f(0)=f^{\mathrm{\prime }}(0)-1=0$.

Let $f(z)$ and $F(z)$ be members of ℋ. The function $f(z)$ is said to be subordinate to $F(z)$, or $F(z)$ is said to be superordinate to $f(z)$, if there exists a function $w(z)$ analytic in $\mathbb{U}$, with $w(0)=0$ and $|w(z)|<1$, and such that $f(z)=F(w(z))$. In such a case, we write $f(z)\prec F(z)$ or $f\prec F$. If the function F is univalent in $\mathbb{U}$, then $f(z)\prec F(z)$ if and only if $f(0)=F(0)$ and $f(\mathbb{U})\subset F(\mathbb{U})$ (cf. [1]).

In particular, the operator ${\mathcal{L}}_2^{\lambda }$ is closely related to the multiplier transformation studied earlier by Flett [3]. Various interesting properties of the operator ${\mathcal{L}}_2^{\lambda }$ have been studied by Jung et al. [4] and Liu [5].

To prove our results, we need the following definitions and theorems considered by Antonimo [6, 7] and Oros [8, 9].

Definition 1.1 ([6], cf. [7, 8])

Let $H(z,\zeta )$ be analytic in $\mathbb{U}\times \overline{\mathbb{U}}$ and let $f(z)$ be analytic and univalent in $\mathbb{U}$. Then the function $H(z,\zeta )$ is said to be strongly subordinate to $f(z)$, or $f(z)$ is said to be strongly superordinate to $H(z,\zeta )$, written as $H(z,\zeta )\prec \prec f(z)$, if for $\zeta \in \overline{\mathbb{U}}$, $H(z,\zeta )$ as the function of z is subordinate to $f(z)$. We note that $H(z,\zeta )\prec \prec f(z)$ if and only if $H(0,\zeta )=f(0)$ and $H(\mathbb{U}\times \overline{\mathbb{U}})\subset f(\mathbb{U})$.

Definition 1.2 ([8], cf. [1])

then $p(z)$ is called a solution of the strong differential subordination. The univalent function $q(z)$ is called a dominant of the solutions of the strong differential subordination, or more simply a dominant, if $p(z)\prec q(z)$ for all $p(z)$ satisfying (1.4). A dominant $\tilde{q}(z)$ that satisfies $\tilde{q}(z)\prec q(z)$ for all dominants $q(z)$ of (1.4) is said to be the best dominant.

Recently, Oros [9] introduced the following strong differential superordinations as the dual concept of strong differential subordinations.

Definition 1.3 ([9], cf. [10])

then $p(z)$ is called a solution of the strong differential superordination. An analytic function $q(z)$ is called a subordinant of the solutions of the strong differential superordination, or more simply a subordinant, if $q(z)\prec p(z)$ for all $p(z)$ satisfying (1.5). A univalent subordinant $\tilde{q}(z)$ that satisfies $q(z)\prec \tilde{q}(z)$ for all subordinants $q(z)$ of (1.5) is said to be the best subordinant.

and are such that $q^{\mathrm{\prime }}(\xi )\ne 0$ for $\xi \in \partial \mathbb{U}\setminus E(q)$. Further, let the subclass of $\mathcal{Q}$ for which $q(0)=a$ be denoted by $\mathcal{Q}(a)$ and $\mathcal{Q}(0)\equiv {\mathcal{Q}}_0$.

Definition 1.4 ([8])

for $z\in \mathbb{U}$, $\xi \in \partial \mathbb{U}\setminus E(q)$, $\zeta \in \overline{\mathbb{U}}$ and $k\ge n$. We write ${\mathrm{\Psi }}_1[\mathrm{\Omega },q]$ as $\mathrm{\Psi }[\mathrm{\Omega },q]$.

Definition 1.5 ([9])

for $z\in \mathbb{U}$, $\xi \in \partial \mathbb{U}$, $\zeta \in \overline{\mathbb{U}}$ and $m\ge n\ge 1$. We write ${\mathrm{\Psi }}_1^{\mathrm{\prime }}[\mathrm{\Omega },q]$ as ${\mathrm{\Psi }}^{\mathrm{\prime }}[\mathrm{\Omega },q]$.

For the above two classes of admissible functions, Oros and Oros proved the following theorems.

Theorem 1.1 ([8])

then $p(z)\prec q(z)$.

Theorem 1.2 ([9])

implies $q(z)\prec p(z)$.

In the present paper, making use of the differential subordination and superordination results of Oros and Oros [8, 9], we determine certain classes of admissible functions and obtain some subordination and superordination implications of multivalent functions associated with the Komatu integral operator ${\mathcal{L}}_c^{\lambda }$ defined by (1.1). Additionally, new differential sandwich-type theorems are obtained. We remark in passing that some interesting developments on differential subordination and superordination for various operators in connection with the Komatu integral operator were obtained by Ali et al. [11–14] and Cho et al. [15].

Firstly, we begin by proving the subordination theorem involving the integral operator ${\mathcal{L}}_c^{\lambda }$ defined by (1.1). For this purpose, we need the following class of admissible functions.

for $z\in \mathbb{U}$, $\zeta \in \partial \mathbb{U}\setminus E(q)$, $\xi \in \overline{\mathbb{U}}$ and $k\ge 1$.

which evidently completes the proof of Theorem 2.1. □

If $\mathrm{\Omega }\ne \mathbb{C}$ is a simply connected domain, then $\mathrm{\Omega }=h(\mathbb{U})$ for some conformal mapping h of $\mathbb{U}$ onto Ω. In this case, the class ${\mathrm{\Phi }}_{\mathcal{L}}[h(\mathbb{U}),q]$ is written as ${\mathrm{\Phi }}_{\mathcal{L}}[h,q]$. The following result is an immediate consequence of Theorem 2.1.

Our next result is an extension of Theorem 2.1 to the case where the behavior of q on $\partial \mathbb{U}$ is not known.

Proof Theorem 2.1 yields ${\mathcal{L}}_c^{\lambda +1}f(z)\prec q_{\rho }(z)$. The result is now deduced from $q_{\rho }(z)\prec q(z)$. □

Proof The proof is similar to that of [[1], Theorem 2.3d] and so is omitted. □

The next theorem yields the best dominant of the differential subordination (2.7).

and q is the best dominant.

Proof Following the same arguments as in [[1], Theorem 2.3e], we deduce that q is a dominant from Theorem 2.2 and Theorem 2.4. Since q satisfies (2.9), it is also a solution of (2.8) and therefore q will be dominated by all dominants. Hence, q is the best dominant. □

In the particular case $q(z)=Mz$, $M>0$, and in view of Definition 2.1, the class of admissible functions ${\mathrm{\Phi }}_{\mathcal{L}}[\mathrm{\Omega },q]$, denoted by ${\mathrm{\Phi }}_{\mathcal{L}}[\mathrm{\Omega },M]$, is described below.

whenever $z\in \mathbb{U}$, $\xi \in \overline{\mathbb{U}}$, $Re\{L{e}^{i\theta }\}\ge (k-1)kM$, $\theta \in \mathbb{R}$ and $k\ge 1$.

In the special case $\mathrm{\Omega }=q(\mathbb{U})=\{w:|w|<M\}$, the class ${\mathrm{\Phi }}_{\mathcal{L}}[\mathrm{\Omega },M]$ is simply denoted by ${\mathrm{\Phi }}_{\mathcal{L}}[M]$.

Proof This follows from Corollary 2.6 by taking $\varphi (u,v,w;z,\xi )=c^{2}w-cv-{(c-1)}^{2}u+C(\xi )$ and $\mathrm{\Omega }=h(\mathbb{U})$, where $h(z)=(c-1)Mz$. To use Corollary 2.6, we need to show that $\varphi \in {\mathrm{\Phi }}_{\mathcal{L}}[\mathrm{\Omega },M]$, that is, the admissible condition (2.10) is satisfied. This follows since

for $z\in \mathbb{U}$, $\xi \in \overline{\mathbb{U}}$, $Re\{L{e}^{-i\theta }\}\ge (k-1)kM$, $\theta \in \mathbb{R}$ and $k\ge 1$. Hence, by Corollary 2.6, we deduce the required results. □

The dual problem of differential subordination, that is, differential superordination of the Komatu integral operator ${\mathcal{L}}_c^{\lambda }$ defined by (1.1), is investigated in this section. For this purpose, the class of admissible functions is given in the following definition.

for $z\in \mathbb{U}$, $\zeta \in \partial \mathbb{U}$, $\xi \in \overline{\mathbb{U}}$ and $m\ge 1$.

which evidently completes the proof of Theorem 3.1. □

If $\mathrm{\Omega }\ne \mathbb{C}$ is a simply connected domain, then $\mathrm{\Omega }=h(\mathbb{U})$ for some conformal mapping h of $\mathbb{U}$ onto Ω. In this case, the class ${\mathrm{\Phi }}_{\mathcal{L}}^{\mathrm{\prime }}[h(\mathbb{U}),q]$ is written as ${\mathrm{\Phi }}_{\mathcal{L}}^{\mathrm{\prime }}[h,q]$. Proceeding similarly as in the previous section, the following result is an immediate consequence of Theorem 3.1.

Theorem 3.1 and Theorem 3.2 can only be used to obtain subordinants of differential superordination of the form (3.1) or (3.2). The following theorem proves the existence of the best subordinant of (3.2) for certain ϕ.

and q is the best subordinant.

Proof The proof is similar to that of Theorem 2.5 and so is omitted. □

Combining Theorem 2.2 and Theorem 3.2, we obtain the following sandwich-type theorem.