# Properties of entire solutions of some linear PDE's

### Andriy Bandura

,### Oleh Skaskiv

,### Petro Filevych

Journal of Applied Mathematics and Computational Mechanics |
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PROPERTIES OF ENTIRE SOLUTIONS OF SOME LINEAR PDE'S

Andriy Bandura^{1}, Oleh Skaskiv^{2}, Petro Filevych^{3}

^{1}Department
of Advanced Mathematics, Ivano-Frankivsk National
Technical University of Oil
and Gas, Ivano-Frankivsk, Ukraine

^{2}Department of Function Theory and Theory of Probability, Ivan
Franko National University of Lviv Lviv, Ukraine

^{3}Department of Information Technologies, Vasyl Stephanyk
Precarpathional National University Ivano-Frankivsk, Ukraine

andriykopanytsia@gmail.com, olskask@gmail.com, filevych@mail.ru

Received: 22 April 2017; accepted: 15 May 2017

**Abstract.** In
this paper, there are improved sufficient conditions of boundedness of the
-index in a direction for entire
solutions of some linear partial differential equations. They are new even for
the one-dimensional case and Also, we found a
positive continuous function such that entire
solutions of the homogeneous linear differential equation with arbitrary fast
growth have a bounded -index and estimated its
growth.

*MSC 2010:** 34M05, 34M10, 35B08, 35B40, 32A15, 32A17*

*Keywords: **linear partial
differential equation, entire function, bounded -index
in direction, bounded -index, homogeneous linear
differential equation, growth of solutions*

1. Introduction

Let be a continuous function. An entire function , , is
called [1-4]* a function of
bounded -index in a direction ,* if there exists such that

(1)

for every and
every , where The
least such integer is called the *-index in the direction of the entire function and is denoted by ** *In the case we
obtain the definition of an entire function of one
variable of bounded -index (see [5,
6]). And the value of the -index
is denoted by

This paper is devoted to three problems in theory of partial differential equations in and differential equations in a complex plane.

At first, we consider the partial differential equation

(2) |

where are entire functions in There are known sufficient conditions [1, 2, 4] of boundedness of the -index in the direction for entire solutions of (2). In particular, some inequalities must be satisfied outside discs of any radius. Replacing the universal quantifier by the existential, we relax the conditions.

Also the ordinary differential equation

(3) |

is considered. Shah, Fricke, Sheremeta, Kuzyk [6-8] did not investigate an index boundedness of the entire solution of (3) because the right hand side of (3) is a function of two variables. But now in view of entire function theory of bounded -index in direction, it is natural to pose and to consider the following question.

**Problem 1** [3,
Problem 4]. *Let * *be
a function of bounded
-index in directions and What
is a function such that an entire solution of equation (3) has a bounded -index?*

Finally, we consider the linear homogeneous differential equation of the form

(4) |

which is obtained from (2), if There is a known result of Kuzyk and Sheremeta [5] about the growth of the entire function of the bounded -index. Later Kuzyk, Sheremeta [6] and Bordulyak [9] investigated the boundedness of the -index of entire solutions of equation (4) and its growth.

Meanwhile, many mathematicians such as
Kinnunen, Heittokangas, Korhonen, Rättya, Cao, Chen, Yang, Hamani,
Belaїdi [10-14] used the
iterated orders to study the growth of solutions (4). Lin, Tu and Shi [15] proposed a more flexible scale to study the growth of solutions. They
used -order. But, the
iterated orders and -orders
do not cover arbitrary growth (see example in [16]). There is considered a more general
approach to describe the relations between the growth of entire coefficients
and entire solutions of (4). In view of results from [16], the authors raise the question: *what is a positive continuous function such
that entire solutions of (4) with arbitrary fast growth have bounded -index?** *We provide an answer to the question.

2. Auxiliary propositions and notations

For and positive continuous function we define By we denote a class of functions which satisfiy the condtion

For simplicity, we also use a notation .

**Theorem A** [1, 4]. *Let An entire in function is
of bounded
-index in direction if and only if there exist numbers and , and such
that for all *

Let us to write , - zeros of the function for a given If for all then we put

**Theorem B** [1,
4]. *Let be an entire function of **the bounded -index in the
direction Then
for every and for every there
exists such that for all *.

**Theorem C** [1,
4]. *Let be an entire function in Then
the function is of bounded -index
in the direction if and only if the following
conditions hold:** **1**) for every there exists such
that for each ** 2**) for every there exists such that for every *

**Theorem D** [1,
4]. *Let . An entire function has** a
bounded
-index in direction if and only if there exist and such
that ** for each **.*

**Theorem E** [17]. *Let be a bounded closed domain in be
a** continuous function, be an entire
function. Then there exists such that for all and for all *

**Theorem F** [5]. *Let be a positive **continuously
differentiable function
of real Suppose that as
where If
an entire function has** a
bounded -index then** *

3. Boundedness of L-index in direction of entire solutions of some linear partial differential equations

Denote where is a zero set of the function The following theorem is valid.

**Theorem 1.** *Let
, and be entire
functions of **the bounded -index
in the direction Suppose that there exist and such
that for each and *

(5) |

*Then an entire
function satisfying *(2)* has bounded -index
in the direction *

**Proof:** Theorem C provides that and Denote Suppose
that Theorem B and
inequality (5) imply that there exist and such that for all

By equation (2), we evaluate the derivative in the direction

The obtained equality implies that for all :

Thus, there exists such that for all

(6) |

If then there exists a sequence of points satisfying (6) and such that with as Substituting in (6) and taking the limit as we obtain that this inequality is valid for all If (i.e. all zeros of belong to ) then by Theorem D the entire function satisfying (2) has a bounded -index in the direction Otherwise, . Since and then there exists such that Let be an arbitrary point from and Since the entire functions have a bounded -index in the direction by Theorem C the set contains at most zeros of the functions or Let be zeros of the slice function (i.e. ) such that where Since we have Obviously,

. |

Thus, if then (6) holds. Hence, for these points the inequality and (6) imply

(8) |

where and

Let be the sum of the diameters of Then Therefore, there exist numbers and such that if then We choose arbitrary points and and connect them by a smooth curve such that and This curve can be selected such that Then on inequality (7) holds. It is easy to prove that the function is continuous on and continuously differentiable except a finite number of points. Moreover, for a complex-valued function of real variable the inequality holds except points, where Then, in view of (7), we have

where Integrating over the variable we deduce i.e. We can choose such that Hence,

(9) |

Since and for all by Cauchy's inequality in variable we obtain

that is (10)

Inequalities (9) and (10) imply that

where Hence,

Denoting we obtain |

Therefore, by Theorem A, the function has a bounded -index in the direction And by Theorem 3 from [1] the function is of the bounded -index in the direction too.

**Remark 1.** *We
require validity of (**5) for
some but nor for all positive Thus, Theorem 1 improves the
corresponding theorem from [1, 4]. The proposition is new even in the
one-dimensional case (see results for the bounded -index
in [6] and bounded index in [8]).*

4. Boundedness of *l*-index of entire
solutions of the equation

We denote

**Theorem 2. ***Let be
an entire function of bounded -index in the
directions for every If
there exist and such
that
* * for all **, then** any entire solution **w(z) of (3) has **a bounded l-index.*

**Proof:** Differentiating (3) in variable and using Theorem B we obtain that for all

. |

Hence, This inequality is
similar to (6). Repeating arguments from Theorem 1, we
deduce that has a bounded *l*-index.
Theorem 2 is proved.

As application of the theorem we consider the differential equation:

(11) |

**Corollary 1.**
*Let be entire function of bounded
-index, Then every entire function satisfying *(11)*
has
**a bounded -index.*

5. The linear homogeneous differential equation with fast growing coefficients

As in [10], let be a strictly increasing positive unbounded function on be an inverse function to We define the order of the growth of an entire function and the function where , is chosen such that And also we need the greater function where is chosen such that Let be the class of positive continuously differentiable on functions such that as We need the following proposition of Bordulyak:

**Theorem G** [9]**.** *Let
and entire functions satisfy the condition for
all If
an entire function is a solution of (4) then is of **the bounded
-index and*

**Theorem 3.** *Let
be a strictly increasing positive
unbounded function on If every entire function has** a bounded
-index ()
then every entire function satisfying *(4)* has **a bounded
-index.** If, in addition, is a
continuously differentiable function of real variable then
*

(12) |

*for every entire transcendental
function satisfying *(4).* *

**Proof:** Since , the following inequalities hold for
arbitrary and It means that Denote
Hence, for one
has

i.e. (5) is valid for By Theorems D and 1 entire solutions of (4) have a bounded -index. It is easy to prove that for all Thus, by Theorem 3 from [1], an entire function satisfying (4) is of the bounded -index, too. The function is a strictly increasing and continuously differentiable function of a real variable. Then Furthermore, as Using Theorem F we obtain (12).

**Theorem 4.** *Let
be a strictly increasing positive
unbounded and continuously differentiable function on If
as then
every entire function satisfying (4) has **a bounded
-index and *

**Proof:**** **At first, we prove that Indeed,

where as As above, one has Hence, Thus, and satisfy conditions of Theorem G with Therefore, every entire function satisfying (4) has a bounded -index.

These theorems are a refinement of results of M. Bordulyak, A. Kuzyk and M. Sheremeta [6, 9]. Unlike these authors, we define the specific function such that entire solutions have a bounded -index. But the function depends of the function Below, we will construct functions and for the entire transcendental function of infinite order.

**Theorem 5.** *For
an arbitrary continuous right differentiable on function
**such that ** ** there
exists a convex on function with the properties** (i) , ; **(ii) for
an unbounded from above set of values .*

**Proof:** For a given we
put and Clearly, the function is continuous on and is fully
contained in a range of this function. For every there exists such
that and for
all . Given the above, it is easy to justify
the existence of increasing to sequence , for which: 1) ; 2) a sequence is increasing to , where for
every ; 3) for all and
every .

Let and for . Clearly, that is a nondecreasing on function. Hence, a function is convex on . For this function we have and for every

i.e. (ii) holds. If for some , then we obtain (i):

This follows from Theorem 5 that , .

**Theorem 6.** *For
an arbitrary entire transcendental function of
infinite order there exists a convex on function
such that **1) ** 2**) for
an unbounded from **the above set of values **
3**) *

**Proof:** We put , .
Since is of infinite order, it follows .
Let be a function constructed for the
function in Theorem 5. Denote . Then . It means that
the function is a convex increasing on half-bounded interval
, where . We
put for and for . By Theorem 5
assumptions 1) and 2) hold. We also obtain . Therefore, 3) is true.

Let where is chosen such that

**Theorem 7.** *For
an arbitrary entire transcendental function of
infinite order there exists a strictly increasing positive unbounded and
continuously differentiable function on with And
if where then *

**Proof:** In view of Theorem 6
we choose where is
an inverse function to Then It is obvious that the function is a strictly increasing positive unbounded and Besides, is a
continuously differentiable function except for the
points of discontinuity of We
estimate a logarithmic derivative of :

where as It implies that .

6. Conclusions

Note that a concept of the bounded -index in a direction has a few advantages in the comparison with traditional approaches to study the properties of entire solutions of differential equations. In particular, if an entire solution has a bounded index, then it immediately yields its growth estimates, a uniform in a some sense distribution of its zeros, a certain regular behavior of the solution, etc.

References

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