**1. Introduction and Motivation**

Let A(*p*) denote the class of functions of the form:

$$f(z) = z^p + \sum\_{n=1}^{\infty} a\_{n+p} z^{n+p} \quad (p \in \mathbb{N} = \{1, 2, 3, \dots\}),\tag{1}$$

which are analytic and *p*-valent in the open unit disk:

$$\mathbb{E} = \{ z : z \in \mathbb{C} \quad \text{and} \quad |z| < 1 \}.$$

In particular, we write:

$$\mathcal{A}(1) = \mathcal{A}\_{\cdot}$$

Furthermore, by S⊂A, we shall denote the class of all functions that are univalent in <sup>E</sup>.

The familiar class of *<sup>p</sup>*-valently starlike functions in <sup>E</sup> will be denoted by <sup>S</sup>∗(*p*), which consists of functions *f* ∈ A(*p*) that satisfy the following conditions:

$$\Re\left(\frac{zf'(z)}{f(z)}\right) > 0 \qquad (\forall \, z \in \mathbb{E})\,.$$

One can easily see that:

$$\mathcal{S}^\*(1) = \mathcal{S}^\*,$$

where S<sup>∗</sup> is the well-known class of normalized starlike functions (see [1]).

We denote by K the class of close-to-convex functions, which consists of functions *f* ∈ A that satisfy the following inequality:

$$\Re\left(\frac{zf'(z)}{g'(z)}\right) > 0 \qquad (\forall \ z \in \mathbb{E})^2$$

for some *g* ∈ S∗.

For two functions *f* and *g* analytic in E, we say that the function *f* is subordinate to the function *g* and write as follows:

$$f \preccurlyeq \mathbf{g} \quad \text{or} \quad f\left(z\right) \preccurlyeq g\left(z\right) \cdot \mathbf{g}$$

if there exists a Schwarz function *w*, which is analytic in E with:

$$w\left(0\right) = 0 \qquad \text{and} \qquad \left|w\left(z\right)\right| < 1\_{\prime}$$

such that:

$$f\left(z\right) = \lg\left(w\left(z\right)\right).$$

Furthermore, if the function *g* is univalent in E, then it follows that:

$$f(z) \preccurlyeq g(z) \quad (z \in \mathbb{E}) \implies f(0) = \emptyset(0) \quad \text{and} \quad f(\mathbb{E}) \subset \operatorname{g}(\mathbb{E}).$$

Next, for a function *f* ∈ A (*p*) given by (1) and another function *g* ∈ A (*p*) given by:

$$g(z) = z^p + \sum\_{n=2}^{\infty} b\_{n+p} z^{n+p} \qquad (\forall \ z \in \mathbb{E})\ \_\times$$

the convolution (or the Hadamard product) of *f* and *g* is given by:

$$(f\*\mathfrak{g})\left(z\right) = z^p + \sum\_{n=2}^{\infty} a\_{n+p} b\_{n+p} z^{n+p} = (\mathfrak{g}\*f)(z).$$

The subclass of <sup>A</sup> consisting of all analytic functions with a positive real part in <sup>E</sup> is denoted by P. An analytic description of P is given by:

$$h(z) = 1 + \sum\_{n=1}^{\infty} c\_n z^n \quad (\forall \ z \in \mathbb{E}).$$

Furthermore, if:

$$\Re\left\{h(z)\right\} > \rho\_{\prime}$$

then we say that *h* is in the class P (*ρ*). Clearly, one see that:

$$\mathcal{P}\left(0\right) = \mathcal{P}\_{\cdot}$$

Historically, in the year 1933, Spaˇcek [2] introduced the *β*-spiral-like functions as follows.

**Definition 1.** *A function f* ∈ A *is said to be in the class* S<sup>∗</sup> (*β*) *if and only if:*

$$\Re\left(e^{i\beta}\frac{zf'(z)}{f'(z)}\right) > 0 \qquad (\forall \, z \in \mathbb{E})$$

*for:*

$$
\beta \in \mathbb{R} \quad \text{and} \quad |\beta| < \frac{\pi}{2},
$$

*where* R *is the set of real numbers.*

In the year 1967, Libera [3] extended this definition to the class of functions, which are spiral-like of order *ρ* denoted by S<sup>∗</sup> *<sup>ρ</sup>* (*β*) as follows.

**Definition 2.** *A function f* ∈A *is said to be in the class* S<sup>∗</sup> *<sup>ρ</sup>* (*β*) *if and only if:*

$$\Re\left(e^{i\beta}\frac{zf'(z)}{f'(z)}\right) > \rho \qquad (\forall \, z \in \mathbb{E})$$

$$\left(0 \le \rho < 1; \ \beta \in \mathbb{R} \quad \text{and} \quad |\beta| < \frac{\pi}{2}\right),$$

*where* R *is the set of real numbers.*

The above function classes S<sup>∗</sup> (*β*) and S<sup>∗</sup> *<sup>ρ</sup>* (*β*) have been studied and generalized by different viewpoints and perspectives. For example, in the year 1974, a subclass *S<sup>α</sup> <sup>β</sup>*(*ρ*) of spiral-like functions was introduced by Silvia (see [4]), who gave some remarkable properties of this function class. Subsequently, Umarani [5] defined and studied another function class *SC*(*α*, *β*) of spiral-like functions. Recently, Noor et al. [6] generalized the works of Silvia [4] and Umarani [5] by defining the class *M*(*p*, *α*, *β*, *ρ*). Here, in this paper, we define certain new subclasses of spiral-like close-to-convex functions by using the idea of Noor et al. [6] and Umarani [5].

We now recall that Kanas et al. (see [7,8]; see also [9]) defined the conic domains Ω*<sup>k</sup>* (*k* 0) as follows:

$$
\Omega\_k = \left\{ u + iv : u > k\sqrt{(u-1)^2 + v^2} \right\}.\tag{2}
$$

By using these conic domains Ω*<sup>k</sup>* (*k* 0), they also introduced and studied the corresponding class *k*-ST of *k*-starlike functions (see Definition 3 below).

Moreover, for fixed *k*, Ω*<sup>k</sup>* represents the conic region bounded successively by the imaginary axis for (*k* = 0), for *k* = 1 a parabola, for 0 < *k* < 1 the right branch of a hyperbola, and for *k* > 1 an ellipse. For these conic regions, the following functions *pk*(*z*), which are given by (3), play the role of extremal functions.

$$\begin{cases} \frac{1+z}{1-z} = 1 + 2z + 2z^2 + \cdots \end{cases} \tag{k=0}$$

$$\begin{cases} \frac{1}{1-z} = 1 + 2z + 2z^2 + \cdots & (k=0) \\\\ 1 + \frac{2}{\pi^2} \left( \log \frac{1+\sqrt{z}}{1-\sqrt{z}} \right)^2 & (k=1) \\\\ \gamma & \text{if } \left( \gamma \right) & \text{if } \gamma \end{cases}$$

$$p\_k(z) = \begin{cases} 1 + \frac{2}{1 - k^2} \sinh^2 \left\{ \left(\frac{2}{\pi} \arccos k\right) \arctan(h\sqrt{z}) \right\} & (0 \le k < 1) \\\\ 1 + \frac{1}{k^2 - 1} \sin \left(\frac{\pi}{2K(\kappa)} \int\_0^{\sqrt{\kappa}} \frac{dt}{\sqrt{1 - t^2} \sqrt{1 - \kappa^2 t^2}}\right) + \frac{1}{k^2 - 1} & (k > 1) \end{cases} \tag{3}$$

$$\left(1+\frac{1}{k^2-1}\sin\left(\frac{\pi}{2K(\kappa)}\int\_0^{\frac{\mu(z)}{\sqrt{\kappa}}} \frac{dt}{\sqrt{1-t^2}\sqrt{1-\kappa^2t^2}}\right)+\frac{1}{k^2-1} \qquad (k>1)\right)$$

where:

$$\mu(z) = \frac{z - \sqrt{\kappa}}{1 - \sqrt{\kappa}z} \qquad \qquad (\forall \ z \in \mathbb{E})$$

and *κ* ∈ (0, 1) is chosen such that:

$$k = \cosh\left(\frac{\pi K'(\kappa)}{4K(\kappa)}\right).$$

Here, *K*(*κ*) is Legendre's complete elliptic integral of the first kind and:

$$K'(\kappa) = K(\sqrt{1 - \kappa^2})\_{\kappa}$$

that is, *K* (*κ*) is the complementary integral of *K* (*κ*).

These conic regions are being studied and generalized by several authors (see, for example, [10–13]). The class *k*-ST is defined as follows.

**Definition 3.** *A function f* ∈ A *is said to be in the class k-*ST *if and only if:*

$$\frac{z f'(z)}{f'(z)} \prec p\_k \left(z\right) \quad \left(\forall \, z \in \mathbb{E}; \, k \ge 0\right)$$

*or, equivalently,*

$$\Re\left(\frac{zf'(z)}{f(z)}\right) > k \left|\frac{zf'(z)}{f(z)} - 1\right|.$$

The class of *k*-uniformly close-to-convex functions denoted by *k*-UK was studied by Acu [14].

**Definition 4.** *A function f* ∈ A *is said to be in the class k-*UK *if and only if:*

$$
\Re\left(\frac{zf'(z)}{g'(z)}\right) > k \left|\frac{zf'(z)}{g'(z)} - 1\right|,
$$

*where g* ∈ *k-*ST *.*

In recent years, several interesting subclasses of analytic functions were introduced and investigated from different viewpoints (see, for example, [6,15–20]; see also [21–25]). Motivated and inspired by the recent and current research in the above-mentioned work, we here introduce and investigate certain new subclasses of analytic and *p*-valent functions by using the concept of conic domains and spiral-like functions as follows.

**Definition 5.** *Let f* ∈ A(*p*). *Then, f* <sup>∈</sup> *k-*K(*p*, *<sup>λ</sup>*) *for a real number <sup>λ</sup> with* <sup>|</sup>*λ*<sup>|</sup> <sup>&</sup>lt; *<sup>π</sup>* <sup>2</sup> *if and only if:*

$$\Re\left(\frac{e^{i\lambda}}{p}\frac{zf'(z)}{\psi(z)}\right) > k\left|\frac{zf'(z)}{\psi(z)} - p\right| + \rho\cos\lambda \quad (k \ge 0; \ 0 \le \rho < 1)$$

*for some ψ* ∈ S∗*.*

**Definition 6.** *Let f* ∈A(*p*). *Then, f* <sup>∈</sup> *k-*Q(*p*, *<sup>λ</sup>*) *for a real <sup>λ</sup> with* <sup>|</sup>*λ*<sup>|</sup> <sup>&</sup>lt; *<sup>π</sup>* <sup>2</sup> *if and only if:*

$$\Re\left(\frac{\epsilon^{i\lambda}}{p}\frac{zf'(z)}{\psi'(z)}\right) > k \left|\frac{(zf'(z))'}{\psi'(z)} - p\right| + \rho \cos\lambda \quad (k \ge 0; \ 0 \le \rho < 1),$$

*for some ψ* ∈ C*.*

**Definition 7.** *Let f* ∈A(*p*) *with:*

$$\frac{f'(z)f'(z)}{pz} \neq 0$$

*and for some real <sup>φ</sup> and <sup>λ</sup> with* <sup>|</sup>*λ*<sup>|</sup> <sup>&</sup>lt; *<sup>π</sup>* <sup>2</sup> *. Then, f* ∈ *k-*Q (*φ*, *λ*, *η*, *f* , *ψ*) *if and only if:*

$$\Re\left(\mathcal{M}\left(\phi,\lambda,\eta,f,\psi\right)\right) > k\left|\mathcal{M}\left(\phi,\lambda,\eta,f,\psi\right) - p\right| + \rho\cos\lambda\_{\prime\prime}$$

*where*

$$\begin{split} \mathcal{M}\left(\phi,\lambda,\eta,f,\psi\right) &= \left(\epsilon^{j\lambda} - \phi\cos\lambda\right) \frac{zf'(z)}{p\psi(z)}\\ &+ \frac{\phi\cos\lambda}{p-\eta} \left(\frac{\left(zf'(z)\right)'}{\Psi'(z)} - \eta\right) \quad \left(\frac{-1}{2} \le \eta < 1\right). \end{split} \tag{4}$$

#### **2. A Set of Lemmas**

Each of the following lemmas will be needed in our present investigation.

**Lemma 1.** (*see [26] p. 70*) *Let h be a convex function in* E *and:*

$$q: \mathbb{E} \implies \mathbb{C} \text{ and } \Re(q\,(z)\,) > 0 \qquad (z \in \mathbb{E})\,.$$

*If p is analytic in* E *with:*

$$p\left(0\right) = h\left(0\right)\_\prime$$

*then:*

$$p\left(z\right) + q\left(z\right)zp'\left(z\right) \prec h\left(z\right) \quad \text{implies} \quad p\left(z\right) \prec h\left(z\right).$$

**Lemma 2.** (*see [26] p. 195*) *Let h be a convex function in* E *with:*

$$h\left(0\right) = 0 \quad \text{and} \quad A > 1.$$

*Suppose that j* <sup>4</sup> *<sup>h</sup>*(0) *and that the functions <sup>B</sup>* (*z*), *<sup>C</sup>* (*z*) *and <sup>D</sup>* (*z*) *are analytic in* <sup>E</sup> *and satisfy the following inequalities*:

$$\Re\left\{B\left(z\right)\right\} \overset{\geqslant}{\geq} A + \left|\mathbb{C}\left(z\right) - 1\right| - \Re\left(\mathbb{C}\left(z\right) - 1\right) + jD\left(z\right), \quad z \in \mathbb{E}.$$

*If p is analytic in* E *with:*

$$p\left(z\right) = 1 + a\_1 z + a\_2 z^2 + \cdots$$

*and the following subordination relation holds true*:

$$A z^2 p''''(z) + B\left(z\right) z p'\left(z\right) + \mathbb{C}\left(z\right) p\left(z\right) + D\left(z\right) \prec h\left(z\right),$$

*then:*

$$p\left(z\right) \prec h\left(z\right).$$
