Let ${ℂ}^n$ denote the space of $n$ complex variables $z=(z_1,\mathrm{\dots },z_n)$ with the Euclidean inner product $⟨z,w⟩={\sum }_{j=1}^n{z}_j{\overline{w}}_j$ and the Euclidean norm $\Vert z\Vert =\sqrt{⟨z,z⟩}$, for all $z,w\in {ℂ}^n$. Also, let ${𝔹}^n$ denote the Euclidean unit ball in ${ℂ}^n$. In the case of one complex variable, the unit disc ${𝔹}^1$ is denoted by $U$.

Let $H({𝔹}^n)$ denote the set of all holomorphic mappings from ${𝔹}^n$ into ${ℂ}^n$. If $f\in H({𝔹}^n)$, we say that $f$ is normalized if $f(0)=0$ and $Df(0)=I_n$, where $Df(z)$ is the complex Jacobian matrix of $f$ at $z$ and $I_n$ is the identity operator in ${ℂ}^n$. Let

be the set of all normalized biholomorphic mappings on ${𝔹}^n$.

A mapping $f\in S({𝔹}^n)$ is called convex (starlike) if its image is a convex (respectively, starlike with respect to the origin) domain in ${ℂ}^n$. We denote by

the class of normalized convex mappings on ${𝔹}^n$ and by

the class of normalized starlike mappings on ${𝔹}^n$. In the case of one complex variable, the sets $S(U)$, $K(U)$ and $S^{*}(U)$ are denoted by $S$, $K$ and $S^{*}$.

If $f\in H({𝔹}^n)$, we say that $f$ is locally biholomorphic on ${𝔹}^n$ if $J_f(z)\ne 0$, for all $z\in {𝔹}^n$, where $J_f(z)=det\left(Df(z)\right)$, for all $z\in {𝔹}^n$. We denote by

the set of all normalized locally biholomorphic mappings on ${𝔹}^n$. If $n=1$, then $ℒ{𝒮}_1({𝔹}^1)$ is denoted by $ℒ𝒮$.

Another important class of normalized holomorphic functions on the unit disc $U$ is the Carathéodory class (for details, one may consult (duren, , Chapter 2), (graham-kohr1, , p. 27) or (pommerenke, , Chapter 2)), denoted by

In the case of several complex variables, we use the family

of normalized holomorphic mappings on the Euclidean unit ball. It is important to mention that the class $ℳ({𝔹}^n)$ plays the role of the Carathéodory family in ${ℂ}^n$. This class will be very important in the section that contains remarks about Loewner chains and Herglotz vector fields. For more details, one may consult graham-hamada-kohr1 , graham-hamada-kohr-kohr1 , graham-kohr1 , kohr1 , pfaltzgraff1 and reich-shoikhet .

Next, we recall the notions of Loewner chain and Herglotz vector field on the Euclidean unit ball in ${ℂ}^n$. We will use these notions to prove that (under some particular assumptions) the convex combination of two Loewner chains is also a Loewner chain.

The following theorem gives a sufficient condition for a mapping $L=L(z,t)$ to be a Loewner chain (see (graham-hamada-kohr1, , Lemma 1.6), graham-kohr-kohr2 , (hamada-kohr2, , Lemma 2.3) or (pfaltzgraff1, , Theorem 2.2)).

The connection between the class $S$ and the Loewner chains in the case of one complex variable is given by a result due to Pommerenke (see e.g. (graham-kohr1, , Theorem 3.1.8)) which says that any function $f\in S$ can be embedded as the first element of a Loewner chain (i.e. for each $f\in S$, there exists a Loewner chain $L(\zeta ,t)$ such that $L(\zeta ,0)=f(\zeta )$, for all $\zeta \in U$).

This result is no longer true for the class $S({𝔹}^n)$, and by this reason I. Graham, H. Hamada and G. Kohr (see graham-hamada-kohr1 ) defined the class

of normalized univalent mappings which have parametric representation on ${𝔹}^n$. It is clear that $S^0({𝔹}^1)=S$ (see pommerenke ), but $S^0({𝔹}^n)⫋S({𝔹}^n)$, for $n\ge 2$ (see graham-hamada-kohr1 and graham-kohr1 ). For details, one may consult also graham-hamada-kohr1 , (graham-kohr1, , Chapter 8) and graham-kohr-kohr2 .

An interesting fact about the class of normalized univalent functions on the unit disc $U$ in $ℂ$ is that the class $S$ is not convex. Namely, starting from two normalized univalent functions on the unit disc even the average of these functions does not necessarily belong to $S$. To show this, we present two examples in the case of one complex variable (see e.g. duren , hallenbeck-macgregor or macgregor ) and one example on the Euclidean unit ball in ${ℂ}^2$ (see e.g. graham-kohr1 or kohr1 ).

In Example 2.1, the functions $f$ and $g$ are not only normalized and univalent - they are even starlike on the unit disc $U$. However, the function $h$ is not starlike on $U$ (in fact, $h$ is not even univalent on $U$). Hence, $f,g\in S^{*}$, but $h\notin S^{*}$ because $h\notin S$. Another important example was given by MacGregor in (macgregor, , Section 3). He proved that the linear combination of two convex functions is not necessarily univalent on the unit disc.

In the previous example, $f,g\in K$ are convex functions, but $h$ is not univalent on $U$ because there exists a point $z_0=(a+i)/(a+1)\in ℂ$ with , where $a=\sqrt{(1-t)/t}$, such that $h^{\prime }(z_0)=0$ (for the complete proof, one may consult (macgregor, , Section 3)).

Next, we can extend the statement of Example 2.1 to the case of several complex variables. For $n=2$, we obtain the following example (considered in (graham-kohr1, , Problem 6.2.3) and (kohr1, , Problem 4.3.4)):

Although linear combinations of univalent functions are not always univalent (for more details about these results, one may consult campbell , hallenbeck-macgregor , hayman or macgregor ), there exist subclasses of the class $S$ that satisfy this condition (see chichra-singh , merkes or trimble in the case $n=1$). The goal of this paper is to extend in the case of several complex variables a result proved by P.N. Chichra and R. Singh in (chichra-singh, , Theorem 2) for the case of one complex variable. This result shows that the convex combination between a starlike function with positive real part of the derivative and the identity function is also starlike on the unit disc $U$, as it follows:

In the case of several complex variables we start with some particular forms of the mapping $f$ in order to construct convex combinations which are starlike on the Euclidean unit ball ${𝔹}^n$. Like in the case of result proved by Chichra and Singh, we also consider convex combinations between a starlike mapping and the identity map in ${ℂ}^n$. First, we prove a general result for normalized locally bihomomorphic mappings on ${𝔹}^n$ which satisfies some additional conditions in order to obtain the starlikeness of the convex combination. Then, in the final part, we propose a conjecture which generalize Theorem 2.5 in the case of several complex variables.

Next we present some important results the will be used in the proofs of the main results from this paper. We recall, without proofs, the analytical characterization of starlikeness in ${ℂ}^n$ (proved by Matsuno, see matsuno or (graham-kohr1, , Theorem 6.2.2); see also elin-reich-shoikhet and suffridge ), the connection between Loewner chains and starlike mappings (see (graham-kohr1, , Corollary 8.2.3) or (pfaltzgraff-suffridge, , Corollary 2)) and also some important criteria for univalence in ${ℂ}^n$.

Using Theorem 3.1 we can prove that a locally biholomorphic mapping with $f(0)=0$ is starlike on the Euclidean unit ball ${𝔹}^n$. Another important characterization of starlikeness was given by Pfaltzgraff and Suffridge (see (graham-kohr1, , Corollary 8.2.3) or (pfaltzgraff-suffridge, , Corollary 2)) in terms of Loewner chains.

We end this section with two important results that ensure the univalence of a normalized holmorphic mapping on the Euclidean unit ball ${𝔹}^n$. The first result, proved by Suffridge in (suffridge, , Theorem 7), is a version of the Noshiro-Warschawski’s univalence criteria (see e.g. (duren, , Theorem 2.16) for the case $n=1$) in the case of several complex variables.

Another important criteria for univalence in ${ℂ}^n$ is presented in the following result proved by I. Graham, H. Hamada and G. Kohr in (graham-hamada-kohr2, , Lemma 2.2) (see also hamada-kohr1 and hamada-kohr2 ).

Taking into account that $S^0({𝔹}^n)⫋S({𝔹}^n)$, for $n\ge 2$ (see graham-hamada-kohr1 ), it is clear that there is an important difference between Theorem 3.3 (which assures us the univalence of a mapping on ${𝔹}^n$) and Theorem 3.4 (which assures us that a mapping admits parametric representation on ${𝔹}^n$), i.e. there exist normalized holomorphic mappings on ${𝔹}^n$ that satisfy condition (3), but do not satisfy condition (4).

Such an example, that shows us that $S^0({𝔹}^n)$ is a proper subclass of $S({𝔹}^n)$ for $n\ge 2$, is presented in (graham-kohr1, , Example 8.3.21).

In view of the results presented in the previous section we can prove some criteria for univalence of a convex combination of normalized holomorphic mappings on the Euclidean unit ball ${𝔹}^n$. In fact, we can obtain a condition for a convex combination to be a mapping which has parametric representation on ${𝔹}^n$.

Let $k\in {\mathbb{N}}^{*}$. We say that a mapping $P_k:{ℂ}^n\to {ℂ}^n$ is a homogenous polynomial of degree $k$ if there exist $Q_k:{ℂ}^n\times \mathrm{\dots }\times {ℂ}^n\to {ℂ}^n$ an $k$-$ℂ$-linear operator such that $P_k(z)=Q_k(z^k)$. For details, one may consult graham-kohr1 or kohr1 . Recall that we define the norm of the operator $P_k$ by $\Vert P_k\Vert =\max \{\Vert P_k(z)\Vert :\Vert z\Vert =1\}$.

In this section we present the first important result of this paper together with some examples to illustrate how this result can be applied in several particular cases. We begin this section with a well-known result related to starlike mappings on the Euclidean unit ball ${𝔹}^n$ (see graham-kohr1 ).

According to Lemma 5.1 we can obtain a first version of Theorem 2.5 in the case of several complex variables. However, in this case we have a particular form of the mapping $f$ (it has on each component a starlike function of one variable). The following result is very simple and its proof is immediate.

It is clear that the mapping $f$ used in the previous result has a very particular form (has on each component a starlike function of one complex variable). However, we can obtain similar results for an arbitrary starlike mapping on the Euclidean unit ball.

In the following examples (considered also in elin-reich-shoikhet , (graham-kohr-kohr1, , Example 3.5), (hamada-kohr1, , Example 3.4), roper-suffridge or (suffridge, , Examples 3 and 7)) we use arbitrary starlike mappings to construct starlike mappings (as convex combinations on ${𝔹}^n$). On the other hand, we can obtain starlike mappings $h$ of the form (5) on ${𝔹}^n$ using starlike mappings $f$ that do not satisfy condition Re$⟨Df(z)(u),u⟩>0$, for all $z\in {𝔹}^n$ and $u\in {ℂ}^n$ with $\Vert u\Vert =1$.

It is important to mention here that the results contained in Examples 5.5 and 5.6 can be directly verified. However, we can obtain starlike mappings $h$ as convex combinations of two starlike mappings such that at least one of them does not satisfy the condition Re$⟨Df(z)(u),u⟩>0$, for all $z\in {𝔹}^n$ and $u\in {ℂ}^n$ with $\Vert u\Vert =1$.

In the following example (considered also in graham-kohr1 and roper-suffridge ) we use a convex mapping $f$ on ${𝔹}^n$ to construct a starlike univalent mapping $h$ on the Euclidean unit ball ${𝔹}^n$. The condition $|a|\le 1/2$ also ensures the univalence of the mapping $h$ on ${𝔹}^n$.