${(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p$-Neighborhood for some classes of multivalent functions

Abstract

In the present paper, we obtain some interesting results for neighborhoods of multivalent functions. Furthermore, we give an application of Miller and Mocanu’s lemma.

MSC:30C45.

1 Introduction and definitions

Let A denote the class of functions f of the form

$$f(z)=z+\sum _{n=2}^{\mathrm{\infty }}{a}_n{z}^n$$

which are analytic in the open unit disk

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

We denote by $A(p,n)$ the class of functions f of the form

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

which are analytic and multivalent in the open unit disk U.

The concept of neighborhood for $f\in A$ was first given by Goodman [1]. The concept of δ-neighborhoods $N_{\delta }(f)$ of analytic functions $f\in A$ was first introduced by Ruscheweyh [2]. Walker [3] defined a neighborhood of analytic functions having positive real part. Owa et al. [4] generalized of the results given by Walker. In 1996, Altıntaş and Owa [5] gave $(n,\delta )$-neighborhoods for functions $f\in A$ with negative coefficients. In 2007, new definitions for neighborhoods of analytic functions $f\in A$ were considered by Orhan et al. [6]. The authors gave the following definition of neighborhoods:

For $f,g\in A$, f is said to be $(\alpha ,\delta )$-neighborhood for g if it satisfies

$$|f^{\prime }(z)-e^{i\alpha }{g}^{\prime }(z)|<\delta (z\in U)$$

for some $-\pi \le \alpha \le \pi$ and $\delta >\sqrt{2(1-cos\alpha )}$. They denote this neighborhood by $(\alpha ,\delta )-N(g)$.

Also, they saw that $f\in (\alpha ,\delta )-M(g)$ if it satisfies

$$|\frac{f(z)}{z}-e^{i\alpha }\frac{g(z)}{z}|<\delta (z\in U)$$

for some $-\pi \le \alpha \le \pi$ and $\delta >\sqrt{2(1-cos\alpha )}$.

In 2009, Altuntaş et al. [7] gave the following definition for neighborhood of analytic functions $f\in A(p,n)$.

For $f,g\in A(p,n)$, f is said to be ${(\alpha ,\delta )}_p$-neighborhood for g if it satisfies

$$|\frac{f^{\prime }(z)}{z^{p-1}}-e^{i\alpha }\frac{g^{\prime }(z)}{z^{p-1}}|<\delta (z\in U)$$

for some $-\pi \le \alpha \le \pi$ and $\delta >p\sqrt{2(1-cos\alpha )}$. They denote this neighborhood by ${(\alpha ,\delta )}_p-N(g)$.

Also, they saw that $f\in {(\alpha ,\delta )}_p-M(g)$ if it satisfies

$$|\frac{f(z)}{z^p}-e^{i\alpha }\frac{g(z)}{z^p}|<\delta (z\in U)$$

for some $-\pi \le \alpha \le \pi$ and $\delta >\sqrt{2(1-cos\alpha )}$.

Recently, Frasin [8] introduced the following definition of $(\alpha ,\beta ,\delta )$-neighborhood for analytic function f in the form

$$f(z)=z-\sum _{n=2}^{\mathrm{\infty }}{a}_n{z}^n(a_n\ge 0).$$

(1.1)

Let f be defined by (1.1). Then f is said to be $(\alpha ,\beta ,\delta )$-neighborhood for $g=z-{\sum }_{n=2}^{\mathrm{\infty }}{b}_n{z}^n$ ($b_n\ge 0$) if it satisfies

$$|e^{i\alpha }{(D^{k}f(z))}^{\prime }-e^{i\beta }{(D^{k}g(z))}^{\prime }|<\delta$$

for some $-\pi \le \alpha ,\beta \le \pi$ and $\delta >\sqrt{2(1-cos(\alpha -\beta ))}$.

The differential operator $D^k$ was introduced by Salagean [9].

Now, we give the following equalities for the functions $f\in A(p,n)$

We define $\mathrm{\wp }:A(p,n)\to A(p,n)$ such that

$$\mathrm{\wp }(f(z))=(\frac{1}{p^{\mathrm{\Omega }}}-\lambda )D^{\mathrm{\Omega }}f(z)+\frac{\lambda }{p}z{(D^{\mathrm{\Omega }}f(z))}^{\prime }(0\le \lambda \le \frac{1}{p^{\mathrm{\Omega }}},\mathrm{\Omega }\in \mathbb{N}\cup \{0\}).$$

(1.2)

We denote by ${\mathrm{\wp }}_{(\mathrm{\Omega },\lambda )}$ the class of analytic functions of the form (1.2) in U.

For $f,g\in {\mathrm{\wp }}_{(\mathrm{\Omega },\lambda )}$, f is said to be ${(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p$-neighborhood for g if it satisfies

$$|e^{i\phi }\frac{{\mathrm{\wp }}^{\mathrm{\prime }}(f(z))}{z^{p-1}}-e^{i\alpha }\frac{{\mathrm{\wp }}^{\mathrm{\prime }}(g(z))}{z^{p-1}}|<\delta (z\in U)$$

for some $-\pi \le \phi -\alpha \le \pi$ and $\delta >p\sqrt{2(1-cos(\phi -\alpha ))}$. We denote this neighborhood by ${(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p-\mathcal{N}(g)$.

Also, we say that $f\in {(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p-\mathcal{M}(g)$ if it satisfies

$$|e^{i\phi }\frac{\mathrm{\wp }(f(z))}{z^p}-e^{i\alpha }\frac{\mathrm{\wp }(g(z))}{z^p}|<\delta (z\in U)$$

for some $-\pi \le \phi -\alpha \le \pi$ and $\delta >\sqrt{2(1-cos(\phi -\alpha ))}$.

We discuss some properties of f belonging to ${(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p-\mathcal{N}(g)$ and ${(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p-\mathcal{M}(g)$.

2 Main results

Theorem 2.1 If $f\in {\mathrm{\wp }}_{(\mathrm{\Omega },\lambda )}$ satisfies

(2.1)

for some $-\pi \le \phi -\alpha \le \pi$ and $\delta >p\sqrt{2(1-cos(\phi -\alpha ))}$, then $f\in {(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p-\mathcal{N}(g)$.

Proof By virtue of (1.2), we can write

If

$$\sum _{k=n}^{\mathrm{\infty }}{\left(\frac{k+p}{p}\right)}^{\mathrm{\Omega }}(k+p)(1+\lambda k{p}^{\mathrm{\Omega }-1})|a_{k+p}-e^{i\alpha }{b}_{k+p}|\le \delta -p\sqrt{2\{1-cos(\phi -\alpha )\},}$$

then we see that

$$|e^{i\phi }\frac{{\mathrm{\wp }}^{\prime }(f(z))}{z^{p-1}}-e^{i\alpha }\frac{{\mathrm{\wp }}^{\prime }(g(z))}{z^{p-1}}|<\delta (z\in U).$$

Thus, $f\in {(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p-\mathcal{N}(g)$. □

Example 2.2

For given

$$g(z)=z^p+\sum _{k=n}^{\mathrm{\infty }}{B}_{k+p}(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })z^{k+p}\in {\mathrm{\wp }}_{(\mathrm{\Omega },\lambda )}(n,p\in \mathbb{N}=\{1,2,\dots \})$$

we consider

$$f(z)=z^p+\sum _{k=n}^{\mathrm{\infty }}{A}_{k+p}(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })z^{k+p}\in {\mathrm{\wp }}_{(\mathrm{\Omega },\lambda )}(n,p\in \mathbb{N}=\{1,2,\dots \})$$

with

$$A_{k+p}=\frac{p^{\mathrm{\Omega }}(\delta -p\sqrt{2(1-cos(\phi -\alpha ))})}{{(k+p)}^{\mathrm{\Omega }+2}(1+\lambda k{p}^{\mathrm{\Omega }-1})(k+p-1)}(n+p-1)e^{-i\phi }+e^{i(\alpha -\phi )}{B}_{k+p}.$$

Then we have that

(2.2)

Finally, in view of the telescopic sum, we can write

$$\begin{array}{rcl}\sum _{k=n}^{\mathrm{\infty }}\frac{1}{(k+p)(k+p-1)}& =& \underset{q\to \mathrm{\infty }}{lim}\sum _{k=n}^q\{\frac{1}{(k+p-1)}-\frac{1}{(k+p)}\}\\ =& \underset{q\to \mathrm{\infty }}{lim}\{\frac{1}{(n+p-1)}-\frac{1}{(p+q)}\}\\ =& \frac{1}{n+p-1}.\end{array}$$

(2.3)

Using (2.3) in (2.2), we have

$$\sum _{k=n}^{\mathrm{\infty }}{\left(\frac{k+p}{p}\right)}^{\mathrm{\Omega }}(k+p)(1+\lambda k{p}^{\mathrm{\Omega }-1})|e^{i\phi }{A}_{k+p}-e^{i\alpha }{B}_{k+p}|=(\delta -p\sqrt{2(1-cos(\phi -\alpha ))}).$$

Therefore, $f\in {(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p-\mathcal{N}(g)$.

Corollary 2.3 If $f\in {\mathrm{\wp }}_{(\mathrm{\Omega },\lambda )}$ satisfies

$$\sum _{k=n}^{\mathrm{\infty }}{\left(\frac{k+p}{p}\right)}^{\mathrm{\Omega }}(k+p)(1+\lambda k{p}^{\mathrm{\Omega }-1})||a_{k+p}|-|b_{k+p}||\le \delta -p\sqrt{2(1-cos(\phi -\alpha ))}$$

for some $-\pi \le \phi -\alpha \le \pi$, $\delta >p\sqrt{2\{1-cos(\phi -\alpha )\}}$, and $arg(a_{k+p})-arg(b_{k+p})=\alpha -\phi$ ($n,p\in \mathbb{N}=\{1,2,\dots \}$), then $f\in {(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p-\mathcal{N}(g)$.

Proof By Theorem 2.1, we see the inequality (2.1) which implies that $f\in {(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p-\mathcal{N}(g)$.

Since $arg(a_{k+p})-arg(b_{k+p})=\alpha -\phi$, if $arg(a_{k+p})={\phi }_{k+p}$, we see $arg(b_{k+p})={\phi }_{k+p}-\alpha +\phi$. Therefore,

$$e^{i\phi }{a}_{k+p}-e^{i\alpha }{b}_{k+p}=e^{i\phi }|a_{k+p}|e^{i{\phi }_{k+p}}-e^{i\alpha }|b_{k+p}|e^{i({\phi }_{k+p}-\alpha +\phi )}=(|a_{k+p}|-|b_{k+p}|)e^{i({\phi }_{k+p}+\phi )}$$

implies that

$$|e^{i\phi }{a}_{k+p}-e^{i\alpha }{b}_{k+p}|=||a_{k+p}|-|b_{k+p}||.$$

(2.4)

Using (2.4) in (2.1), the proof of the corollary is complete. □

Theorem 2.4 If $f\in {\mathrm{\wp }}_{(\mathrm{\Omega },\lambda )}$ satisfies

$$\sum _{k=n}^{\mathrm{\infty }}{\left(\frac{k+p}{p}\right)}^{\mathrm{\Omega }}(1+\lambda k{p}^{\mathrm{\Omega }-1})|e^{i\phi }{a}_{k+p}-e^{i\alpha }{b}_{k+p}|\le \delta -\sqrt{2(1-cos(\alpha -\phi ))}$$

for some $-\pi \le \phi -\alpha \le \pi$ and $\delta >\sqrt{2\{1-cos(\phi -\alpha )\}}$, then $f\in {(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p-\mathcal{M}(g)$.

The proof of this theorem is similar with Theorem 2.1.

Corollary 2.5 If $f\in {\mathrm{\wp }}_{(\mathrm{\Omega },\lambda )}$ satisfies

$$\sum _{k=n}^{\mathrm{\infty }}{\left(\frac{k+p}{p}\right)}^{\mathrm{\Omega }}(1+\lambda k{p}^{\mathrm{\Omega }-1})||a_{k+p}|-|b_{k+p}||\le \delta -\sqrt{2(1-cos(\phi -\alpha ))}$$

for some $-\pi \le \phi -\alpha \le \pi$, $\delta >\sqrt{2\{1-cos(\phi -\alpha )\}}$, and $arg(a_{k+p})-arg(b_{k+p})=\alpha -\phi$, then $f\in {(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p-\mathcal{M}(g)$.

Next, we derive the following theorem.

Theorem 2.6 If $f\in {(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p-\mathcal{N}(g)$, $0\le \phi <\alpha \le \pi$ and $arg(e^{i\phi }{a}_{k+p}-e^{i\alpha }{b}_{k+p})=k\phi$, then

$$\sum _{k=n}^{\mathrm{\infty }}{\left(\frac{k+p}{p}\right)}^{\mathrm{\Omega }}(k+p)(1+\lambda k{p}^{\mathrm{\Omega }-1})|e^{i\phi }{a}_{k+p}-e^{i\alpha }{b}_{k+p}|\le \delta -p\{cos\phi -cos\alpha \}.$$

Proof For $f\in {(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p-\mathcal{N}(g)$, we have

Let us consider z such that $argz=-\phi$. Then $z^k=|z{|}^k{e}^{-ik\phi }$. For such a point $z\in U$, we see that

This implies that

$${\{\sum _{k=n}^{\mathrm{\infty }}{\left(\frac{k+p}{p}\right)}^{\mathrm{\Omega }}(k+p)(1+\lambda k{p}^{\mathrm{\Omega }-1})|e^{i\phi }{a}_{k+p}-e^{i\alpha }{b}_{k+p}||z{|}^k+p(cos\phi -cos\alpha )\}}^2<{\delta }^2$$

or

$$p(cos\phi -cos\alpha )+\sum _{k=n}^{\mathrm{\infty }}{\left(\frac{k+p}{p}\right)}^{\mathrm{\Omega }}(k+p)(1+\lambda k{p}^{\mathrm{\Omega }-1})|e^{i\phi }{a}_{k+p}-e^{i\alpha }{b}_{k+p}||z{|}^k<\delta$$

for $z\in U$. Letting $|z|\to 1^{-}$, we have that

$$\sum _{k=n}^{\mathrm{\infty }}{\left(\frac{k+p}{p}\right)}^{\mathrm{\Omega }}(k+p)(1+\lambda k{p}^{\mathrm{\Omega }-1})|e^{i\phi }{a}_{k+p}-e^{i\alpha }{b}_{k+p}|\le \delta -p(cos\phi -cos\alpha ).$$

 □

Theorem 2.7 $f\in {(\phi ,\alpha ,\delta ,\lambda ,\mathrm{\Omega })}_p-\mathcal{M}(g)$, $0\le \phi <\alpha \le \pi$ and $arg(e^{i\phi }{a}_{k+p}-e^{i\alpha }{b}_{k+p})=k\phi$, then

$$\sum _{k=n}^{\mathrm{\infty }}{\left(\frac{k+p}{p}\right)}^{\mathrm{\Omega }}(1+\lambda k{p}^{\mathrm{\Omega }-1})|e^{i\phi }{a}_{k+p}-e^{i\alpha }{b}_{k+p}|\le \delta +cos\alpha -cos\phi .$$

The proof of this theorem is similar with Theorem 2.6.

Remark 2.8 Taking $\phi =0$, $\mathrm{\Omega }=0$, $\lambda =0$ and $p=1$ in Theorem 2.6, we obtain the following theorem due to Orhan et al. [6].

Theorem 2.9 If $f\in (\alpha ,\delta )-\mathcal{N}(g)$ and $arg(a_n-e^{i\alpha }{b}_n)=(n-1)\phi$ ($n=2,3,4,\dots$), then

$$\sum _{n=2}^{\mathrm{\infty }}n|a_n-e^{i\alpha }{b}_n|\le \delta +cos\alpha -1.$$

Remark 2.10 Taking $\phi =0$, $\mathrm{\Omega }=0$ and $\lambda =0$ in Theorem 2.6, we obtain the following theorem due to Altuntaş et al. [7].

Theorem 2.11 If $f\in {(\alpha ,\delta )}_p-\mathcal{N}(g)$ and $arg(a_{k+p}-e^{i\alpha }{b}_{k+p})=k\phi$, then

$$\sum _{k=n}^{\mathrm{\infty }}(k+p)|a_{k+p}-e^{i\alpha }{b}_{k+p}|\le \delta -p(1-cos\alpha ).$$

We give an application of following lemma due to Miller and Mocanu [10].

Lemma 2.12 Let the function

$$w(z)=b_n{z}^n+b_{n+1}{z}^{n+1}+b_{n+2}{z}^{n+2}+\cdots (n\in \mathbb{N})$$

be regular in the unit disk U with $w(z)\not\equiv 0$ ($z\in U$). If $z_0=r_0{e}^{i{\theta }_0}$ ($r_0<1$) and $|w(z_0)|={max}_{|z|\le r_0}|w(z)|$, then $z_0{w}^{\prime }(z_0)=mw(z_0)$ where m is real and $m\ge n\ge 1$.

Theorem 2.13 If $f\in {\mathrm{\wp }}_{(\mathrm{\Omega },\lambda )}$ satisfies

$$|e^{i\phi }\frac{{\mathrm{\wp }}^{\prime }(f(z))}{z^{p-1}}-e^{i\alpha }\frac{{\mathrm{\wp }}^{\prime }(g(z))}{z^{p-1}}|<\delta (p+n)-p\sqrt{2(1-cos(\phi -\alpha ))}$$

for some $-\pi \le \phi -\alpha \le \pi$ and $\delta >(\frac{p}{p+n})\sqrt{2(1-cos(\phi -\alpha ))}$, then

$$|e^{i\phi }\frac{\mathrm{\wp }(f(z))}{z^p}-e^{i\alpha }\frac{\mathrm{\wp }(g(z))}{z^p}|<\delta +\sqrt{2(1-cos(\phi -\alpha ))}(z\in U).$$

Proof Let us define $w(z)$ by

$$e^{i\phi }\frac{\mathrm{\wp }(f(z))}{z^p}-e^{i\alpha }\frac{\mathrm{\wp }(g(z))}{z^p}=e^{i\phi }-e^{i\alpha }+\delta w(z).$$

(2.5)

Then $w(z)$ is analytic in U and $w(0)=0$. By logarithmic differentiation, we obtain from (2.5) that

$$\frac{e^{i\phi }{\mathrm{\wp }}^{\prime }(f(z))-e^{i\alpha }{\mathrm{\wp }}^{\prime }(g(z))}{e^{i\phi }\mathrm{\wp }(f(z))-e^{i\alpha }\mathrm{\wp }(g(z))}-\frac{p}{z}=\frac{\delta w^{\prime }(z)}{e^{i\phi }-e^{i\alpha }+\delta w(z)}.$$

Since

$$\frac{e^{i\phi }{\mathrm{\wp }}^{\prime }(f(z))-e^{i\alpha }{\mathrm{\wp }}^{\prime }(g(z))}{z^p(e^{i\phi }-e^{i\alpha }+\delta w(z))}=\frac{p}{z}+\frac{\delta w^{\prime }(z)}{e^{i\phi }-e^{i\alpha }+\delta w(z)},$$

we see that

$$e^{i\phi }\frac{{\mathrm{\wp }}^{\prime }(f(z))}{z^{p-1}}-e^{i\alpha }\frac{{\mathrm{\wp }}^{\prime }(g(z))}{z^{p-1}}=p(e^{i\phi }-e^{i\alpha })+\delta w(z)(p+\frac{z{w}^{\prime }(z)}{w(z)}).$$

This implies that

$$|e^{i\phi }\frac{{\mathrm{\wp }}^{\prime }(f(z))}{z^{p-1}}-e^{i\alpha }\frac{{\mathrm{\wp }}^{\prime }(g(z))}{z^{p-1}}|=|p(e^{i\phi }-e^{i\alpha })+\delta w(z)(p+\frac{z{w}^{\prime }(z)}{w(z)})|.$$

We claim that

$$|e^{i\phi }\frac{{\mathrm{\wp }}^{\prime }(f(z))}{z^{p-1}}-e^{i\alpha }\frac{{\mathrm{\wp }}^{\prime }(g(z))}{z^{p-1}}|<\delta (p+n)-p\sqrt{2(1-cos(\phi -\alpha ))}$$

in U.

Otherwise, there exists a point $z_0\in U$ such that $z_0{w}^{\prime }(z_0)=mw(z_0)$ (by Miller and Mocanu’s lemma) where $w(z_0)=e^{i\theta }$ and $m\ge n\ge 1$.

Therefore, we obtain that

$$\begin{array}{rcl}|e^{i\phi }\frac{{\mathrm{\wp }}^{\prime }(f(z_0))}{z_0^{p-1}}-e^{i\alpha }\frac{{\mathrm{\wp }}^{\prime }(g(z_0))}{z_0^{p-1}}|& =& |p(e^{i\phi }-e^{i\alpha })+\delta e^{i\theta }(p+m)|\\ \ge & \delta (p+m)-\left|p(e^{i\phi }-e^{i\alpha })\right|\\ \ge & \delta (p+n)-p\sqrt{2(1-cos(\phi -\alpha )).}\end{array}$$

This contradicts our condition in Theorem 2.13. □

Hence, there is no $z_0\in U$ such that $|w(z_0)|=1$. This implies that $|w(z)|<1$ for all $z\in U$. Thus, we have that

$$\begin{array}{rcl}|e^{i\phi }\frac{\mathrm{\wp }(f(z))}{z^p}-e^{i\alpha }\frac{\mathrm{\wp }(g(z))}{z^p}|& =& |(e^{i\phi }-e^{i\alpha })+\delta w(z)|\\ \le & |e^{i\phi }-e^{i\alpha }|+\delta |w(z)|\\ <& \delta +\sqrt{2(1-cos(\phi -\alpha ))}.\end{array}$$

Letting $\phi =0$, $\mathrm{\Omega }=0$, $\lambda =0$ and $\alpha =\frac{\pi }{2}$ in Theorem 2.13, we can obtain the following corollary.

Corollary 2.14 If $f\in A(p,n)$ satisfies

$$|\frac{f^{\prime }(z)}{z^{p-1}}-i\frac{g^{\prime }(z)}{z^{p-1}}|<\delta (p+n)-p\sqrt{2}(z\in U)$$

for some $\delta >\sqrt{2}(\frac{p}{p+n})$, then

$$|\frac{f(z)}{z^p}-i\frac{g(z)}{z^p}|<\delta +\sqrt{2}(z\in U).$$