Starlikeness, convexity and uni- formly close-to-convexity of the modified Lommel function are also discussed

https://doi.org/10.5831/HMJ.2020.42.4.719

CERTAIN GEOMETRIC PROPERTIES OF MODIFIED LOMMEL FUNCTIONS

Muhey U Din^∗ and Sibel Yalc¸in

Abstract. In this article, we find some sufficient conditions under which the modified Lommel function is close-to-convex with respect to − log(1 − z) and ¹₂log

1+z 1−z



. Starlikeness, convexity and uni- formly close-to-convexity of the modified Lommel function are also discussed. Some results related to the H. Silverman are also the part of our investigation.

1. Introduction and preliminaries

The Lommel function of the first kind s_κ,η(z) is a particular solution of the in-homogeneous Bessel differential equation

(1) z²w⁰⁰(z) + zw⁰(z) + (z²− η²)w(z) = z^κ+1, and it can be expressed in terms of hypergeometric series

sκ,η(z) = z^κ+1

(κ − η + 1) (κ + η + 1) ¹F2



1;κ − η + 3

2 ,κ + η + 3 2 ; −z²

4

 , where κ ± η is not negative odd integer. It is observed that, Lommel function sκ,η does not belong to the class A. Thus, the normalized Lom- mel function of first kind is defined as:

L_κ,η(z) = (κ − η + 1) (κ + η + 1) z^1−κ2 sκ,η(√ z) (2)

L_κ,η(z) =

∞

X

n=0

−1 4

n

(B)_n(V)_nzⁿ⁺¹, (3)

Received April 18, 2020. Revised October 19, 2020. Accepted October 21, 2020.

2010 Mathematics Subject Classification. 30C45, 33C10, 30C20, 30C75.

Key words and phrases. Analytic functions, Modified Lommel fiuctions, Close- to-convexity, Starlikeness.

*Corresponding author

where B = ^κ−η+3₂ , V = ^κ+η+3₂ and (a)_n shows the Appell symbol which is defined in terms of Eulers gamma functions such that (a)_n= ^Γ(a+n)_Γ(a) = a(a + 1)...(a + n − 1). Clearly the function Lκ,η belongs to the class A.

To discuss the certain properties of normalized Lommel functions with different aspects, here we define modified form of the normalized Lommel functions

L(z) = z

1 + z ∗ L_κ,η(z) =

∞

X

n=0

1

4ⁿ(B)_n(V)_nzⁿ⁺¹

= z +

∞

X

n=2

1

4ⁿ⁻¹(B)_n−1(V)_n−1zⁿ. (4)

Let A be the class of functions of the form

(5) f (z) = z +

∞

X

n=2

anzⁿ,

analytic in the open unit disc U = {z : |z| < 1} and satisfy the normal- ization condition f (0) = f⁰(0) − 1 = 0. Let S^∗(α) , C (α) and K (α) denote the classes of starlike, convex and close-to-convex functions of order α respectively and are defined as:

S^∗(α) =



f : f ∈ A and < zf⁰(z) f (z)



> α, z ∈ U , α ∈ [0, 1)

 , C (α) =



f : f ∈ A and < (zf⁰(z))⁰ f⁰(z)



> α, z ∈ U , α ∈ [0, 1)



and K (α) =



f : f ∈ A and < zf⁰(z) g (z)



> α,

z ∈ U , α ∈ [0, 1) , g ∈ S^∗(0) :≡ S^∗

 . Recently, many researchers studied the geometric properties of some special functions with different approaches. Special functions have great importance in pure and applied mathematics. The wide use of these functions have attracted many researchers to work on the different di- rections. For details we refer here [2, 3, 7, 10, 14, 15]. Certain conditions for close-to-convexity of some special functions like Bessel functions, q- Mittag-Leffler functions, Wright functions, Dini functions have deter- mined by many mathematicians with different methods (for details see

[2, 4, 11, 12, 13]). To prove our main results, we need the following lemmas:

Lemma 1.1. [20] If the function f (z) = z + a₂z²+ ... + a_nzⁿ+ ...

is analytic in U and in addition 1 ≥ 2a₂ ≥ ... ≥ na_n ≥ ... ≥ 0 or 1 ≤ 2a2 ≤ ... ≤ na_n... ≤ 2, then f (z) is close-to-convex function with respect to the convex function z → −Log (1 − z) .

Lemma 1.2. [20] If the odd function g(z) = z + b₃z³ + ... + b2n−1z²ⁿ⁻¹+... is analytic in U and if 1 ≥ 3b3 ≥ ... ≥ (2n+1)b_2n+1... ≥ 0 or 1 ≤ 3b₃≤ ... ≤ (2n + 1)b_2n+1... ≤ 2, then g(z) is univalent in U .

We can verify directly that if a function f : U → C satisfies the hypothesis of Lemma 1.1, then it is close-to-convex with respect to the convex function

z 7→ 1

2log 1 + z 1 − z

 .

Lemma 1.3. [18] A function f defined in (5) belongs to the classes S^∗(α) and C (α) , if it satisfies the following conditions

∞

X

n=2

(n − α) |an| ≤ 1 − α, α ∈ [0, 1) ,

∞

X

n=2

n (n − α) |an| ≤ 1 − α, α ∈ [0, 1) respectively.

Lemma 1.4. [6] If f ∈ A satisfy |f⁰(z) − 1| < 1 for each z ∈ U , then f is convex in U_1/2=z : |z| < ¹₂ .

Lemma 1.5. [8] If f ∈ A satisfy   

f (z) z − 1

< 1 for each z ∈ U , then f is starlike in U_1/2=z : |z| < ¹₂ .

Lemma 1.6. [9] If f ∈ A satisfy |f⁰(z) − 1| < ^√2

5 for each z ∈ U , then f is starlike in U_1/2=z : |z| < ¹₂ .

Lemma 1.7. [16] Let β ∈ C with <(β) > 0, c ∈ C with |c| ≤ 1, c 6= −1. If h ∈ A satisfies

c |z|^2β +



1 − |z|^2β

zh⁰⁰(z) βh⁰(z)    

≤ 1, z ∈ U ,

then the integral operator

Cβ(z) =





 β

z

Z

0

t^β−1h⁰(t)dt







1/β

, z ∈ U , is analytic and univalent in U .

Lemma 1.8. [19] If f ∈ A satisfies   

zf⁰⁰(z) f⁰(z)

< ¹₂, then f ∈ U CV.

Lemma 1.9. [5] Let {a_n}^∞_n=1 be a sequence of non negative real numbers such that a1 = 1. If {an}^∞_n=2 is convex decreasing. i.e. 0 ≥ an+2− a_n+1 ≥ a_n+1− a_n, then

<

( _∞ X

n=1

a_nzⁿ⁻¹ )

> 1

2, (z ∈ U ) .

2. Close to Convexity of Modified Lommel Functions with Respect to Certain Functions

In this section we will discuss some conditions on the parameters µ and v under which the modified Lommel functions are assured Close-to- Convex with respect to the functions

− log(1 − z) and 1

2log 1 + z 1 − z

 . By using the Lemma 1.1, we will get the following results.

Theorem 2.1. If κ, η ∈ R⁺ and κ ≥ η with inequality

B ≥ 3

8 (V + 1) − 1,

then z → L(z) is close-to-convex with respect to convex function

− log (1 − z) . Proof. Set

L(z) = z +

∞

X

n=2

a_n−1zⁿ.

Here an−1 = ₄n−1(B)_n−1¹ (V)_n−1, we have an−1 > 0 for all n ≥ 2 and a₁ = _4BV¹ < 1.

From (4) , we have Ωa_n= na_n−1− (n + 1)a_n

= 4n (B + n − 1) (V + n − 1) 4ⁿ(B)_n(V)_n



1 − n + 1

4n (B + n − 1) (V + n − 1)



= 4n (B + n − 1) (V + n − 1) 4ⁿ(B)_n(V)_n κ(n), where

κ(n) = 1 − n + 1

4n (B + n − 1) (V + n − 1).

In view of Lemma 1.1, we have to show that κ(n) ≥ 0. By using the inequality 4n ≥ n + 1, ∀n ≥ 1 it is observed that κ(n) ≥ 0, since (B + n − 1) ≥ 0 and (V + n − 1) ≥ 0, ∀n ≥ 2. Thus L(z) is close-to- convex with respect to convex function − log (1 − z) with B ≥ _8(V+1)³ − 1.

Theorem 2.2. If κ, η ∈ R⁺ and κ ≥ η with inequality

B ≥ 5

12 (V + 1) − 1,

then z → L(z) is close-to-convex with respect to convex function

1 2log

1+z 1−z

 . Proof. Set

f (z) = zL(z²) = z +

∞

X

n=2

A_2n−1z²ⁿ⁻¹.

Here A2n−1 = an−1 = ₄n−1(B)_n−1¹ (V)_n−1, therefore we have a1 = a1 =

1

4BV ≤ 1 and A_2n−1 > 0 for all n ≥ 2. To prove our main result we will prove that {(2n − 1)an−1}_n≥2 is a decreasing sequence. By a short computation we obtain

Ωa_n = (2n − 1)a_n−1− (2n + 1)a_n

= 4 (2n − 1) (B + n − 1) (V + n − 1) 4ⁿ(B)_n(V)_n



1 − 2n + 1

4 (2n − 1) (B + n − 1) (V + n − 1)



= 4 (2n − 1) (B + n − 1) (V + n − 1) 4ⁿ(B)_n(V)_n φ (n) ,

where

φ (n) = 1 − 2n + 1

4 (2n − 1) (B + n − 1) (V + n − 1).

In view of Lemma 1.2, we have to show that φ (n) ≥ 0. By using the inequality 4 (2n − 1) ≥ 2n + 1, ∀n ≥ 1 it is observed that φ (n) ≥ 0,

∀n ≥ 2. Thus L(z) is close-to-convex with respect to convex function

1 2log

1+z 1−z



with B ≥ _12(V+1)⁵ − 1.

3. Sufficient Conditions Involving Results of H. Silverman In this section, we are interested to find out the conditions on the pa- rameters of modified lommel functions, under these conditions L defined in (4) belongs to the various subclasses of starlike and convex func- tions. For similar results involving Gaussian hypergeometric functions and Bessel functions, we refer [1, 17].

Theorem 3.1. If κ, η ∈ R⁺, α ∈ [0, 1) and κ ≥ η with inequality Φ {Φ (2 − α) + α − 1} ≤ 4ΦBV (Φ − 1) (1 − α) ,

where, Φ = (B + 1) (V + 1) , then L defined in (4) belongs to the class S^∗(α) .

Proof. Set

L(z) = z +

∞

X

n=2

a_n−1zⁿ,

here an−1 = ₄n−1(B)_n−1¹ (V)_n−1,we have an−1 > 0 for all n ≥ 2 and a1 =

1

4BV < 1. From Lemma 1.3 we need only show that

∞

X

n=2

(n − α) |a_n| ≤ 1 − α.

Consider (6)∞

X

n=2

(n − α) a_n−1 =

∞

X

n=2

n − 1

4ⁿ⁻¹(B)_n−1(V)_n−1 +

∞

X

n=2

1 − α

4ⁿ⁻¹(B)_n−1(V)_n−1.

By using the expression (Ψ)_n−1 = Ψ (Ψ + 1)_n−2, we may write (6) as follows

∞

X

n=2

(n − α) a_n−1

= 1 4

∞

X

n=2

4 (n − 1)

4ⁿ⁻¹B (B+1)_n−2V (V+1)_n−2 (7)

+

∞

X

n=2

1 − α

4ⁿ⁻¹B (B+1)_n−2V (V+1)_n−2. By using the inequalities

4ⁿ⁻¹ ≥ 4 (n − 1) , n ∈ N, (Ψ + 1)_n−1 ≥ (Ψ + 1)ⁿ⁻¹, n ∈ N the expression (7) becomes

∞

X

n=2

(n − α) an−1 ≤ 1 4BV

 ∞

X

n=2

 1

(B+1) (V+1)

n−2

+ (1 − α)

∞

X

n=2

 1

4 (B+1) (V+1)

n−2

= Φ

4BV

 1

Φ − 1 +1 − α Φ

 .

This sum is bounded above by 1 − α if and only if Φ {Φ (2 − α) + α − 1}

≤ 4ΦBV (Φ − 1) (1 − α) holds.

Theorem 3.2. If κ, η ∈ R⁺, α ∈ [0, 1) and κ ≥ η with inequality Φ (2 − α) ≤ 2BV (1 − α) (Φ − 1) ,

where, Φ = (B + 1) (V + 1) , then L defined in (4) belongs to the class C (α) .

Proof. Set

L(z) = z +

∞

X

n=2

a_n−1zⁿ,

here a_n−1 = ₄n−1(B)_n−1¹ (V)_n−1,we have a_n−1 > 0 for all n ≥ 2. From Lemma 1.3 we need only show that

∞

X

n=2

n (n − α) |a_n| ≤ 1 − α.

Consider (8)

∞

X

n=2

n (n − α) a_n−1=

∞

X

n=0

(n + 2) (n + 2 − α) a_n+1 = ρ − ασ, where

ρ =

∞

X

n=0

(n + 2)²a_n+1 and σ =

∞

X

n=0

(n + 2) a_n+1.

Thus, we have to calculate the conditions under which the ρ−ασ ≤ 1−α.

For this by using the inequality 4ⁿ⁺¹ ≥ (n + 2)², ∀ n ≥ 0 and the expression (Ψ)_n−1= Ψ (Ψ + 1)_n−2 we obtain

ρ ≤ 1

BV

∞

X

n=0

 1

(B+1) (V+1)

n

= (B+1) (V+1) BV {(B+1) (V+1) − 1}. (9)

Analogously, by using the inequality 4ⁿ⁺¹≥ 2 (n + 2) , ∀ n ≥ 0 and the expression (Ψ)_n−1= Ψ (Ψ + 1)_n−2 we obtain

σ ≤ 1

2BV

∞

X

n=0

 1

(B+1) (V+1)

n

= (B+1) (V+1)

2BV {(B+1) (V+1) − 1}. (10)

Therefore the expression ρ − ασ with the help of (9) and (10) becomes ρ − ασ ≤ (B+1) (V+1)

BV {(B+1) (V+1) − 1}− (B+1) (V+1) 2BV {(B+1) (V+1) − 1}, is bounded above by 1−α if and only if Φ (2 − α) ≤ 2BV (1 − α) (Φ − 1) , where, Φ = (B + 1) (V + 1) holds.

Theorem 3.3. If κ, η ∈ R⁺ and κ ≥ η with inequalities 4 (B+n − 2) (V+n − 2) ≥ 1,

(B+n − 2) (V+n − 2) (B+n − 1) (V+n − 1) + 1 ≥ 8 (B+n − 1) (V+n − 1) , then

<

L(z) z



> 1

2, ∀ z ∈ U .

Proof. To prove this, we have to show that (11) {a_n}^∞_n=1=

 1

4ⁿ⁻¹(B)_n−1(V)_n−1

∞ n=1

is a convex decreasing sequence. For this a_n− a_n+1 = 1

4ⁿ⁻¹

 1

B (B+1)_n−2V (V+1)_n−2

− 1

4B (B+1)_n−1V (V+1)_n−1



= 1

4ⁿ⁻¹BV

 4 (B+n − 2) (V+n − 2) − 1 4 (B+1)_n−1(V+1)_n−1



> 0,

where ∀ m ≥ 1, B ≥³₂, V ≥³₂ and 4 (B+n − 2) (V+n − 2) ≥ 1. Now, to show that the sequence defined in (11) is convex decreasing, we prove that a_n− 2a_n+1+ a_n+2≥ 0.

Take

a_n− 2a_n+1+ a_n+2

= 1

4ⁿ⁻¹(B)_n−1(V)_n−1 − 2

4ⁿ(B)_n(V)_n + 1

4ⁿ⁺¹(B)_n+1(V)_n+1

= 1

4ⁿ⁻¹BV











(B+n − 2) (V+n − 2) (B+n − 1) (V+n − 1)

−8 (B+n − 1) (V+n − 1) + 1 16 (B+1)_n(V+1)_n











> 0.

The above expression is non-negative ∀ m ≥ 1, B ≥³₂, V ≥³₂ and (B+n − 2) (V+n − 2) (B+n − 1) (V+n − 1) + 1

≥ 8 (B+n − 1) (V+n − 1) , which shows that the sequence defined in (11) is decreasing and convex.

From Lemma 1.9

<

∞

X

n=1

anzⁿ⁻¹

!

> 1/2, ∀ z ∈ U , which is equivalent to

<

L(z) z



> 1

2, ∀ z ∈ U .

4. Starlikeness and Convexity of Modified Lommel Func- tions in U_1/2

Theorem 4.1. Let κ, η ∈ R⁺ and κ ≥ η. Then the following asser- tions are true:

(i) If 4(B+1)(V+1)−1^(B+1)(V+1) < BV, then Lκ,η is starlike in U_1/2. (ii) If (B+1)(V+1)−1^(B+1)(V+1) < 2BV, then Lκ,η is convex in U_1/2.

Proof. (i) By using the well-known triangle inequality and the in- equality (B)_n≥ (B)ⁿ, for all n ∈ N, we obtain

Lκ,η(z) z − 1

≤     

∞

X

n=1

1

4ⁿ(B)_n(V)_nzⁿ     

≤ 1

4BV

∞

X

n=1

 1

4 (B+1) (V+1)

n−1

≤ (B+1) (V+1) BV {4 (B+1) (V+1) − 1}.

In view of Lemma 1.5, Lκ,η is starlike in U_1/2, if BV{4(B+1)(V+1)−1}^(B+1)(V+1) < 1, but this is true under the hypothesis.

(ii) Conisder, 

L⁰κ,η(z) − 1 ≤

∞

X

n=1

n + 1 4ⁿ(B)_n(V)_n.

Since, 4ⁿ ≥ 2 (n + 1) , for all n ∈ N and (B)_n ≥ (B)ⁿ, for all n ∈ N.

Therefore 

L⁰_κ,η(z) − 1

 ≤ 1

2BV

∞

X

n=1

 1

(B+1) (V+1)

n−1

= (B+1) (V+1)

2BV {(B+1) (V+1) − 1}.

In view of Lemma 1.4, Lκ,η is convex in U_1/2, if 2BV{(B+1)(V+1)−1}^(B+1)(V+1) < 1, but this is true under the hypothesis.

Theorem 4.2. Let α ∈ [0, 1) , κ, η ∈ R⁺ and κ ≥ η, G =4 (B+1) (V+1) = (κ + 5)²− η²

and H =4BV = (κ + 3)²− η². Then for all z ∈ U the following assertions are true:

(i) If κ > −5 +p

2 + η² and (G−2)(GH−G−H)^G(G−1) < 1 − α, then Lκ,η ∈ S^∗(α) .

(ii) If κ > −5 +p

3 + η² and (G−3)(2GH−4G−3H)^2G(2G−3) < 1 − α, then Lκ,η ∈ C^∗(α) .

Proof. The proof of the above Theorem is similar as Theorem 2.1 in [21].

Consider the integral operator F_β : U → C, where β ∈ C, β 6= 0,

F_β(z) =





 β

z

Z

0

t^β−2Lκ,η(t) dt







1 β

, z ∈ U .

Here F_β ∈ A. In the next theorem, we obtain the conditions so that F_β is univalent in U .

Theorem 4.3. Let κ, η ∈ R⁺and κ ≥ η. Let (G−2) (GH − G − H) >

G (G−1) and suppose that M is a positive real number such that

|Lκ,η(z)| ≤ M in the open unit disc. If

|β − 1| + G (G−1)

(G−2) (GH − G − H)+ M

|β| ≤ 1, then Fβ is univalent in U .

Proof. A calculations gives us zF_β⁰⁰(z)

F_β⁰(z) = zL⁰κ,η(z)

Lκ,η(z) +z^β−1

β Lκ,η(z) + β − 2, z ∈ U .

Since Lκ,η ∈ A, then by the Schwarz Lemma, triangle inequality and Theorem (4.2) , we obtain



1 − |z|²

     

zF_β⁰⁰(z) F_β⁰(z)

≤ 

1 − |z|² (

|β − 1| +    

zL⁰κ,η(z) Lκ,η(z) − 1

+|z|^R(β)

|β|

Lκ,η(z) z

)

≤ 

1 − |z|²



|β − 1| + G (G−1)

(G−2) (GH − G − H) + M

|β|



≤ 1.

This shows that the given integral operator satisfies the Becker’s crite- rion for univalence, hence F_β is univalent in U .

5. Uniformly Convexity of Modified Lommel Functions Theorem 5.1. Let κ, η ∈ R⁺ and κ ≥ η. If (G−3) (2GH−4G−3H) >

4G (2G−3) , then Lκ,η ∈ U CV.

Proof. Since from Theorem 4.2 

zL⁰⁰_κ,η(z) L⁰_κ,η(z)

≤ 2G (2G−3)

(G−3) (2GH−4G−3H). By using Lemma 1.8, we have

zL⁰⁰κ,η(z) L⁰_κ,η(z)

< 1 2, if

2G (2G−3)

(G−3) (2GH−4G−3H) < 1 2. This implies that

(G−3) (2GH−4G−3H) > 4G (2G−3) . Hence the required result.

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Muhey U Din

Department of Mathematics,

Government Post Graduate Islamia College Faisalabad, Pakistan.

E-mail: muheyudin@yahoo.com Sibel Yal¸cin

Department of Mathematics,

Faculty of Arts and Science, Bursa Uludag University, Bursa, Turkey.

E-mail: syalcin@uludag.edu.tr.