Let’s consider the function *f* defined by
$$f(z)=\frac{1}{{z}^{p}}{\left(1-\lambda +\lambda \frac{\mathrm{sin}z}{z}\right)}^{\alpha -p},\phantom{\rule{thickmathspace}{0ex}}z\in \mathcal{U},$$(15)

where 0 ≤ *α* < *p*, the power is the principal one, and assuming that the parameter λ ∈ ℂ is chosen such that
$$\frac{1}{\lambda}\ne 1-\frac{\mathrm{sin}z}{z},\phantom{\rule{thickmathspace}{0ex}}z\in \mathcal{U}.$$(16)

Using MAPLE™ software, from Figure 1a we may see that
$$\underset{\left|z\right|\le 1}{max}\left|1-\frac{\mathrm{sin}z}{z}\right|<0.18,$$(17)

therefore (16) holds whenever $\left|\lambda \right|\le \frac{50}{9}=\mathrm{5.555...}$, and consequently, if λ ∈ ℂ satisfies this inequality then *f* ∈ Σ_{p,2}.

Using again MAPLE™ software, from Figure 1b we have that
$$\underset{\left|z\right|\le 1}{max}\left|{\mathrm{sin}}^{2}\frac{z}{2}\right|<0.28,$$(18)

and a simple computation leads to
$$\left|{\left[{z}^{p}f(z)\right]}^{\frac{1}{\alpha -p}}\left(\frac{z{f}^{\prime}(z)}{f(z)}+\alpha \right)+p-\alpha \right|=2(p-\alpha )\left|\lambda \right|\left|{\mathrm{sin}}^{2}\frac{z}{2}\right|<2(p-\alpha )\left|\lambda \right|\cdot 0.28,\phantom{\rule{thickmathspace}{0ex}}z\in \mathcal{U}.$$

Thus, according to Theorem 2.2 we obtain the following special case:

**Example 3.1:** *If* λ ∈ ℂ *and* $\left|\lambda \right|\le \frac{75}{14\sqrt{10}}=\mathrm{1.6941...}$, *then*
$$f(z)=\frac{1}{{z}^{p}}{\left(1-\lambda +\lambda \frac{\mathrm{sin}z}{z}\right)}^{\alpha -p}\in \mathrm{\Sigma}{S}_{p,2}^{\ast}(\alpha ),$$*for some real values of α* (0 ≤ *α* < *p*), where the power is the principal one.

As we already proved, if $\left|\lambda \right|\le \frac{50}{9}=\mathrm{5.555...}$ then the relation (16) holds. Therefore, there exists a function *f* ∈ Σ_{p,2} such that
$${f}^{\prime}(z)=-\frac{p}{{z}^{p+1}}{\left(1-\lambda +\lambda \frac{\mathrm{sin}z}{z}\right)}^{\alpha -p},z\in \mathcal{U},$$

where 0 ≤ *α* < *p*, and the power is the principal one, assuming that λ ∈ ℂ is chosen such that $\left|\lambda \right|\le \frac{50}{9}=\mathrm{5.555...}$

A simple computation combined with (18) shows that
$$\left|{\left(\frac{{z}^{p+1}{f}^{\prime}(z)}{-p}\right)}^{\frac{1}{a-p}}\left(1+\frac{z{f}^{\u2033}(z)}{{f}^{\prime}(z)}+\alpha \right)+p-\alpha \right|=2(p-\alpha )\left|\lambda \right|\left|{\mathrm{sin}}^{2}\frac{z}{2}\right|<2(p-\alpha )\left|\lambda \right|\cdot 0.28,\phantom{\rule{mediummathspace}{0ex}}z\in \mathcal{U},$$

and from Theorem 2.4 we obtain the following special case:

**Example 3.3:** *If* λ ∈ ℂ *and* $\left|\lambda \right|\le \frac{75}{14\sqrt{10}}=\mathrm{1.6941...}$, *then f* ∈ Σ_{p,2} *with*
$${f}^{\prime}(z)=-\frac{p}{{z}^{p+1}}{\left(1-\lambda +\lambda \frac{\mathrm{sin}z}{z}\right)}^{a-p},z\in \mathcal{U},$$(19)*for some real values of α* (0 ≤ *α* < *p*), *is in* Σ*K*_{p, n}(α). *(The power is the principal one)*.

As we proved at the beginning of this section, the function *f* ∈ Σ_{p,2}, where
$$f(z)=\frac{1}{{z}^{p}}{\left(1-\lambda +\lambda \frac{\mathrm{sin}z}{z}\right)}^{\frac{1}{\gamma}},\phantom{\rule{mediummathspace}{0ex}}z\in \mathcal{U},$$(21)

with λ ∈ ℂ, $\left|\lambda \right|\le \frac{50}{9}=\mathrm{5.555...}$, and the power is the principal one. Using the inequality (20), we deduce that
$$\left|\frac{\gamma {[{z}^{p}f(z)]}^{\gamma}}{z}\left(\frac{z{f}^{\prime}(z)}{f(z)}+p\right)\right|=\left|\lambda \right|\left|\frac{z\mathrm{cos}z-\mathrm{sin}z}{{z}^{2}}\right|<\left|\lambda \right|\cdot 0.38,\phantom{\rule{mediummathspace}{0ex}}z\in \mathcal{U},$$

and from Theorem 2.7 we obtain the next special case:

**Example 3.5:** *If* λ ∈ ℂ *and* $\left|\lambda \right|\le \frac{75}{19\sqrt{10}}=\mathrm{1.2483...}$, *then the function f given by* (21) *is in* $\mathrm{\Sigma}{S}_{p,2}^{\ast}\left(p+\frac{1}{\gamma}\right)$. *for* $\gamma \le -\frac{1}{p}$. *(The power is the principal one)*.

Using MAPLE™ software, we could check that the next inequalities hold (see Figures 3a, 3b, 4a, and 4b):
$$\underset{\left|z\right|\le 1}{max}\left|1+\frac{z}{2}-\frac{{e}^{z}-1}{z}\right|<0.22,$$(22)
$$\underset{\left|z\right|\le 1}{max}\left|{e}^{z}-1-z\right|<0.73,$$(23)
$$\underset{\left|z\right|\le 1}{max}\left|{e}^{z}-1\right|<1.73,$$(24)
$$\underset{\left|z\right|\le 1}{max}\left|\frac{z{e}^{z}-{e}^{z}+1}{{z}^{2}}-\frac{1}{2}\right|<\mathrm{0.51.}$$(25)

From (22) and (23), using Theorem 2.2 we may easily obtain the following special case:

**Example 3.6:** *If* λ ∈ ℂ *and* $\left|\lambda \right|\le \frac{300}{73\sqrt{10}}=\mathrm{1.2996...}$, *then*
$$f(z)=\frac{1}{{z}^{p}}{\left[1+\lambda \left(\frac{{e}^{z}-1}{z}-1-\frac{z}{2}\right)\right]}^{\alpha -p}\in \mathrm{\Sigma}{S}_{p,2}^{\ast}(\alpha ),$$*for some real values of α* (0 ≤ *α* < *p*), *where the power is the principal one*.

From (22) and (24), using Theorem 2.4 we easily get the next special case:

**Example 3.8:** *If* λ ∈ ℂ *and* $\left|\lambda \right|\le \frac{300}{73\sqrt{10}}=\mathrm{1.2996...}$, *then f* ∈ Σ_{p,2} *with*
$${f}^{\prime}(z)=-\frac{p}{{z}^{p+1}}{\left[1+\lambda \left(\frac{{e}^{z}-1}{z}-1-\frac{z}{2}\right)\right]}^{\alpha -p},\phantom{\rule{mediummathspace}{0ex}}z\in \mathcal{U},$$(26)*for some real values of α (0 ≤ α < p), is in* Σ*K*_{p}, n(*α*). (*The power is the principal one*).

Finally, from the inequalities (22) and (25), using Theorem 2.7 we obtain the next special case:

**Example 3.10:** *If* λ ∈ ℂ *and* $\left|\lambda \right|\le \frac{150}{51\sqrt{10}}=\mathrm{0.93008...}$, *then*
$$f(z)=-\frac{1}{{z}^{p}}{\left[1+\lambda \left(\frac{{e}^{z}-1}{z}-1-\frac{z}{2}\right)\right]}^{\frac{1}{\gamma}},\in \mathrm{\Sigma}{S}_{p,2}^{\ast}\left(p+\frac{1}{\gamma}\right),$$*for* $\gamma \le -\frac{1}{p}$. *(The power is the principal one)*.

We will omit the detailed proofs of the last three examples, since these are similar with the previous ones.

## Comments (0)