On a Certain Subclass of Multivalent Functions. A Research Submitted by. Muntadher Ali Joudah. Supervised by Dr. Najah Ali Al -Ziadi

Transcription

1 Republic of Iraq Ministry of Higher Education and Scientific Research University of Al-Qadisiyah / College of Education Department of Mathematics On a Certain Subclass of Multivalent Functions A Research Submitted by Muntadher Ali Joudah To the council of the department of mathematics /collage of education, University of Al-Qadisiyah in Partial fulfillment of the requirements for bachelor in mathematics Supervised by Dr. Najah Ali Al -Ziadi 2018 A.C A.H.

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3 الأهداء الى النور الذي ينير لي درب النجاح ويامن علماني الصمود مهما تبدلت الظروف... أبي وأمي الى كل من علمني حرفا... أساتذتي الأعزاء وبالخصوص أستاذي العزيز الدكتور نجاح علي الزيادي الذي ساعدني ووقف معي في العمل

4 Abstract We presented in this work a certain class MA(ʎ,α,σ,β,m,p) of multivalent analytic functions with linear operator in the open unit disk U. We study coefficient inequality, distortion and growth theorems, radii of starlikeness, convexity and close - to - convexity, weighted mean and arithmetic mean, extreme points. I

5 CONTAINS Subject Page Chapter One 1.1 Basic Definitions Standard Results 9 Chapter Two 2.1 Introduction Coefficient bounds Distortion and growth theorems Radii of starlikeness, convexity and 15 close-to-convexity 2.5 Weighted mean and arithmetic mean Extreme points 20 References 23 II

6 Chapter One Basic Definitions and Standard Results

7 Chapter One Basic Definitions Chapter One Basic Definitions and Standard Results Introduction: In this chapter, we list out all the definitions of the family of functions from analytic, univalent and multivalent ( valent) and all related terms used during the investigation. We also include in this chapter all the standard theorems and lemmas used in the work. 1.1 Basic Definitions Definition (1.1.1)[5]: A function of the complex variable is analytic at a point if its derivative exists not only at but each point in some neighborhoods of. Itis analytic in region if it is analytic at every point in. Definition (1.1.2)[5]: A function is said to be univalent (schilcht) if it does not take the same value twice i.e. for all pairs of distinct points,. In other words, is one to one (or injective) mapping of onto another domain. If assumes the same value more than one, then is said to be multivalent (valent) in. We also deal with the functions which are meromorphic univalent in the punctured unit disk = { C: < < }. is said to be meromorphic if it is analytic at every point in except finite elements in. As examples, the function = is univalent in while = is not univalent in. Also, = + is univalent in for each positive integer n. Example (1.1.1 ) [5]:The function = + is univalent in U. Let, and suppose =. Then + = + 1

8 Chapter One Basic Definitions + + = = + + =. Since, <, we know that + +. Hence = or = Definition ( 1.1.3) [5]: A function is said to be locally univalent at a point Cif it is univalent in some neighborhood of. For analytic function, the condition is equivalent to local univalent at. Example (1.1.2)[5]: Consider the domain = { C: < <, < arg < }, and the function: Cgiven by =. It is clear thatis analytic onand local univalent at every point, since for all. However,is not univalent on, since ( + ) = ( ) =. Definition (1.1.4)[5]: Let denotes the class of functions of the form: = +, N. which are analytic and univalent in the open unit disk. Definition (1.1.5)[5]: We say that is normalized if satisfies the conditions = and =. Definition (1.1.6)[5]: A set C is said to be starlike with respect to if the linear segment joining to every other point lies entirely in. In a more picturesque language, the requirement is that every point of is visible = 2

9 Chapter One Basic Definitions from. The set is said to be convex if it is starlike with respect to each of its points, that is, if the linear segment joining any two points of lies entirely in. Definition (1.1.7)[5]: A function is said to be conformal at a point if it preserves the angle between oriented curves passing through in magnitude as well as in sense. Geometrically, images of any two oriented curves taken with their corresponding orientations make the same angle of intersection as the curves at both in magnitude and direction. A function = is said to be conformal in the domain, if it is conformal at each point of the domain. Definition (1.1.8)[5]: A function is said to be starlike function of order if and only if { } >, < ;,.. Denotes the class of all starlike functions of order in by and the class of all starlike functions of order, =. Geometrically, we can say that a starlike function is conformal mapping of the unit disk onto a domain starlike with respect to the origin. For example, the function is starlike function of order. =, Definition (1.1.9)[5]: A function is said to be convex function of order if and only if { + } >, < ;,.. Denotes the class of all convex functions of order in by and for the convex function =. 3

10 Chapter One Basic Definitions Definition (1.1.10)[5]: A function is said to be close to convex of order < if there is a convex function such that { } >, ;.. We denote by, the class of close to convex functions of order, is normalized by the usual conditions = =. By using argument, we can write the condition (1.4) as <, >,.. We note that. Definition(1.1.11)[6]:A Möbius transformation, or a bilinear transformation, is a rational function : C C of the form = where,,, Care fixed and. + +, Example(1.1.3)[5]:Perhaps the most important member of is the Koebe function which is given by = = + + +, and maps the unit disk to the complement of the ray (, ]. This can be verified by writing = + ( ), 4

11 Chapter One Basic Definitions and noting that + maps the unit disk conformally onto the right half- plane {{} > }; see Fig.... Fig. (1.1.1): The Koebe function maps conformally onto C\ (, ]. We note that = +, =, =. Now = + ( ) =. And Möbius transformation that maps onto the right half plane whose boundary is the imaginary axis. Also, is the squaring function, while translates the image one space to the left and then multiplies it by a factor of. Note that the Koebe function is starlike, but not convex. Definition (1.1.12)[5]: Let denote the class of analytic valently functions in of the form: = +,, N = {,, }.. =+ We say that is valently starlike of order, valently convex of order, and valently close - to - convex of order <, respectively if and only if : { } >. 5

12 Chapter One Basic Definitions { + } > { } >... Definition (1.1.13)[5]: Let us denote by (p) the class of meromorphic function of the form: = +, N. = which are meromorphic and valent in the punctured unit disk = { C: < < } = {}. We say that is valently meromorphic starlike of order < if and only if { } > for.. Also, is valently meromorphic convex of order < if and only if { + } >,.. Note that if =, we have defined univalent meromorphic starlike of order <, univalent meromorphic convex of order < respectively. Denoted by (1) the class of univalent and meromorphic functions in. Definition (1.1.14)[5]: Radius of starlikeness of a function is the largest, < < for which it is starlike in <. Definition (1.1.15)[5]: Radius of convexity of a function is the largest, < < for which it is convex in <. 6

13 Chapter One Basic Definitions Definition (1.1.16)[9]:The convolution (or Hadamard product) of the functionsand denoted by is defined as following for the functions in and respectively: (i) If then (ii) If then = +, = +, =+ =+ = +. =+ = +, = +, = = = +. and if = in (i), then the convolution (or Hadamard product) for the functions in. Also, if = in (ii), then the convolution (or Hadamard product) for the functions in (1). Definition (1.1.17)[5]:The weighted mean of and defined by = = [ + + ], < <. Also, h =, = is the arithmetic mean of =,,,,. 7

14 Chapter One Basic Definitions Definition (1.1.18)[8]: Let be a topological vector space over the field C and let be a subset of. A point is called an extreme point of if it has no representation of the form = +, < < as a proper convex combination of two distinct points and in. 8

15 Chapter One Standard Results 1.2 Standard Results The following lemmas and theorems are essential and has been used in the proofs of the our principal results in the next chapter. Lemma (1.2.1)[3]: Let. Then, > if and only if + < +, where be any complex number. Theorem (1.2.1)[5]: (Distortion Theorem) For each + +, = <.. For each, equality occurs if and only if is a suitable rotation of the Koebe function. We say upper and lower bounds for as Distortion bounds. Theorem (1.2.2)[5]: (Growth Theorem) For each, = <. +. For each, equality occurs if and only if is a suitable rotation of the Koebe function. Theorem (1.2.3)[5]: (Maximum Modulus Theorem) Suppose that a function is continuous on boundary of ( any disk or region). Then, the maximum value of, which is always reached, occurs somewhere on the boundary of and never in the interior. 9

16 Chapter Two On a Certain Subclass of Multivalent Functions

17 Chapter two On a Certain Subclass of Multivalent Functions 2.1: Introduction Let A(p) indicate the class of functions of the form: fz = Z + a Z Z U, β N = {,,. },. =+ Which are analytic and multivalent in the open unit disk U = {Z C: Z < } Let M denote the subclass of A(p) containing of function of the form: fz = Z a Z a, β N = {,,. },. =+ Which are analytic and multivalent in the open unit disk U. Definition (2.1)[7]: let σ,, m N, σ,, m, β N and Then, we wefine the linear operator D σ,, Aβ Aβ by fz = Z + a k Z k. k=+ D σ,, fz = Z + + =+ Z U.. n βσ β + a Z, With the help of the integral operator we define the class MAʎ,, σ,, m, β. Definition (2.1): A function f M is said to be in the class MAʎ,, σ,, m, β if and only if Re { D σ,, fz / + z D σ,, fz // σ, fz / + ʎz D σ, //},., fz D, 10

18 Chapter two On a Certain Subclass of Multivalent Functions Where β N, α < ʎ ʎ+, σ, α and m Some of the following properties studied for other classes in [,,]. 2.2: Coefficient bounds The following theorem gives a necessary and sufficient condition for function to be in the class MAʎ,, σ,, m, β. Theorem (2.1): Let fz M. Then fz MAʎ,, σ,, m, β if and only if =+ + Where β N, < n βσ β + (n ʎn ʎn + n)a β ʎβ ʎβ + β,. ʎ ʎ+ The result is sharp for the function fz = z, σ, and m. β ʎβ ʎβ + β + σ + (n ʎn ʎn + n) N. Proof: Assume that fz MAʎ,, σ,, m, β, so we have Re { D σ,, D, β z Re { ʎβ ʎβ + βz or equivalently fz / + z D, fz // σ, //} fz σ, fz / + ʎz D, =+ =+ n + σ + a z z, n β + ; β + σ + ʎn ʎn + nz } 11

19 Chapter two On a Certain Subclass of Multivalent Functions (β ʎβ ʎβ + β)z + σ =+ + (n ʎn ʎn + n)a z Re { ʎβ ʎβ + βz + σ =+ + ʎn ʎn + nz. This inequality is correct for all z U. letting z yields Re {(β ʎβ ʎβ + β) Therefore =+ + =+. + n βσ β + (n ʎn ʎn + n)a } n βσ β + (n ʎn ʎn + n)a β ʎβ ʎβ + β. Conversely, let (2.5) hold. We will prove that (2.4) is correct and then fz MAʎ,, σ,, m, β. By lemma (1.2.1) it is enough, show that w β + < w + β where W = D σ,, fz / + z D σ,, fz // D σ,, fz / + ʎz D σ,, fz // or show that T = Nz D σ,, fz / + z D σ,, fz // β + D σ,, fz / β + D σ,, fz // 12

20 Chapter two On a Certain Subclass of Multivalent Functions < Nz D σ,, fz / + z D σ,, fz // + β D σ,, fz / + β ʎz D σ,, fz // = Q, where Nz = D σ,, fz / + ʎz D σ,, fz // and it is easy to verify that Q T > and so the proof is complete. Finally, sharpness follows if we take fz = z β ʎβ ʎβ + β + σ + (n ʎn ʎn + n) N. Corollary (2.1): Let f MAʎ,, σ,, m, β. Then a β ʎβ ʎβ + β + σ + (n ʎn ʎn + n) N.. 2.3: Distortion and growth theorems z, n β + ; β, n β + ; β We introduce here the distortion and growth theorems for the functions in the class MAʎ,, σ,, m, β. Theorem (2.2): Let the function fz defined by (2.2) be in the class MAʎ,, σ,, m, β. Then, for z = r < < r (β ʎβ ʎβ + β)r + + σ + (β + ʎβ + ʎβ + + β + ) r fz (β ʎβ ʎβ + β)r + +,. + σ + β + (ʎβ + ʎβ + + β + ) for z U. The result (2.8) is sharp. 13

21 Chapter two On a Certain Subclass of Multivalent Functions Proof: Since fz MAʎ,, σ,, m, β, in view of theorem (2.1), we have ( + σ β + ) β + (ʎβ + ʎβ + + β + ) + =+ which immediately yields =+ a =+ a n βσ β + (n ʎn ʎn + n)a β ʎβ ʎβ + β, β ʎβ ʎβ + β + σ + β + (ʎβ + ʎβ + + β + ) Consequently, for z = r < <, we obtain fz r + r + a =+ r (β ʎβ ʎβ + β)r σ + β + (ʎβ + ʎβ + + β + ) and fz r r + a =+ r (β ʎβ ʎβ + β)r + + σ + β + (ʎβ + ʎβ + + β + ) This completes the proof of theorem (2.2). Finally, by taking the function. 14

22 Chapter two On a Certain Subclass of Multivalent Functions fz = z ʎ ʎ+ + σ p+β + (ʎ+ ʎ+++) we can show that the result of theorem (2.2) is sharp. Theorem (2.3): If fz MAʎ,, σ,, m, β, then z +,. β βr + (β ʎβ ʎβ + β)r + σ + β + (ʎβ + ʎβ + + β + ) βr f / z β + (β ʎβ ʎβ + β)r +. + σ + β + (ʎβ + ʎβ + + β + ) The result is sharp for the function f is given by (2.9) Proof: The proof is similar to that of theorem (2.2). 2.4: Radii of starlikeness, convexity and close to - convexity Using the inequalities (1.7), (1.8), (1.9) and theorem (2.1), we can compute the radii of starlikeness, convexity and close to convexity. Theorem (2.4): Let fz MAʎ,, σ,, m, β. Then fz is p valently starlike of order ρ ρ < in the disk z < R, where R = inf { ρ+ pσ p+β ʎ ʎ+ ρ( ʎ ʎ+) } p The result is sharp for the function fz given by (2.6). Proof: It is sufficient to show that for z < R, we have zf / z fz β β ρ ρ <,, n β + ; β N. 15

23 Chapter two On a Certain Subclass of Multivalent Functions Thus if zf / z fz β =+ n βa =+ a z. =+ zf / z fz n ρ β ρ a β β ρ, z. Hence, by Theorem (2.1), (2.10) will be true if and hence n ρ β ρ z 16 z. + σ + (n ʎn ʎn + ) β ʎβ ʎβ + β β ρ + σ + (n ʎn ʎn + ) z { } n ρ(β ʎβ ʎβ + β) β + ; β N. Setting z = R, we get the desired result. p Theorem (2.5): Let fz MAʎ,, σ,, m, β. Then f is p valently convex of order ρ ρ < in the disk z < R, where β ρ + σ + n ʎ n + ʎ R = inf { } n ρβ ʎ β + ʎ β + ; β N. The result is sharp with the extermal function f given by (2.6). Proof: It is sufficient to show that + zf // z f / z for z < R, we have β β ρ ρ <, p, n, n

24 Chapter two On a Certain Subclass of Multivalent Functions Thus if + zf // z f / z =+ =+ β nn βa β na + zf // z f / z nn ρ ββ ρ a 17 z =+ z β β ρ, z. Hence, by Theorem (2.1), (2.11) will be true if and hance n ρ β ρ z. + σ + n ʎ n + ʎ β ʎ β + ʎ β ρ + σ + n ʎ n + ʎ z { } n ρβ ʎ β + ʎ N. Setting z = R, we get the desired result. p,, n β + ; β Theorem (2.6): Let a function fz MAʎ,, σ,, m, β. Then f is p valently close to convex of order ρ ρ < in the disk z < R, where β ρ + σ + n ʎ n + ʎ R = inf { } ββ ʎ β + ʎ β + ; β N. p The result is sharp, with the extermal function fz given by (2.6). Proof: It is sufficient to show that f / z β β ρ ρ <, z, n

25 Chapter two On a Certain Subclass of Multivalent Functions for z < R, we have that Thus if f / z z β na z =+ =+ f / z β β ρ, z na z β ρ Hence, by Theorem (2.1), (2.12) will be true if... and hence β ρ z + σ + n ʎ n + ʎ ββ ʎ β + ʎ, β ρ + σ + n ʎ n + ʎ z { } ββ ʎ β + ʎ N. Setting z = R, we get the desired result. p, n β + ; β 2.5: Weighted mean and arithmetic mean Theorem (2.7): Let f and g be in the class MAʎ,, σ,, m, β. Then the weighted mean of f and g is also in the class MAʎ,, σ,, m, β. Proof: By Definition (1.1.17), we have E z = [ qfz + + qgz] = [ q z a z + + q z b z ] =+ 18 =+

26 Chapter two On a Certain Subclass of Multivalent Functions = z [ qa + + qb ]z. =+ Since f and g are in the class MAʎ,, σ,, m, β so by Theorem (2.1), we get and =+ =+ + + n βσ β + (n ʎn ʎn + n)a β ʎβ ʎβ + β n βσ β + (n ʎn ʎn + n)b β ʎβ ʎβ + β, hence =+ + n βσ β qb ) = n βσ q + β + =+ + + q + =+ (n ʎn ʎn + n) ( qa n βσ β + q(β ʎβ ʎβ + β) (n ʎn ʎn + n) a (n ʎn ʎn + n) b + + q(β ʎβ ʎβ + β) = (β ʎβ ʎβ + β). 19

27 Chapter two On a Certain Subclass of Multivalent Functions This shows E MAʎ,, σ,, m, β. In the following theorem, we shall prove that the class MAʎ,, σ,, m, β is closed under arithmetic mean. Theorem (2.8): let f z, f z,, f s z defined by f k z = z =+ a z, (a,k, k =,,, s, n β + ). be in the class MAʎ,, σ,, m, β. Then the arithmetic mean of f k z k =,,, s defined by hz = s f kz, is also in the class MAʎ,, σ,, m, β. 20 S k= Proof: by (2.13) and (2.14), we can write S hz = s z k= =+ a,k z = z = S s a,kz. Since f k z MAʎ,, σ,, m, β for every k =,,., s, Sα by using theorem (2.1), we prove that =+ + n βσ β + = n βσ + s β + k= S =+ k= (n ʎn ʎn + n) s a,k S k= (n ʎn ʎn + n) a,k s (β ʎβ ʎβ + β) = β ʎβ ʎβ + β. k= This ends the proof of theorem (2.8). 2.6: Extreme points In the following theorem, we obtain the extreme points of the class MAʎ,, σ,, m, β.

28 Chapter two On a Certain Subclass of Multivalent Functions Theorem (2.9): Let f z = z and f z = z where β N, < β ʎβ ʎβ + β + σ + (n ʎn ʎn + n) ʎ ʎ+, σ, and m. z,. Then the function f is in the class MAʎ,, σ,, m, β if and only if it can be expressed in the form: fz = θ z + θ f z =+,. where (θ, θ, n β + )and θ + =+ θ =. Proof: Suppose that f is expressed in the form (2.16). then fz = θ z + θ [z =+ β ʎβ ʎβ + β + σ + (n ʎn ʎn + n) z ] hance = z =+ β ʎβ ʎβ + β + σ + (n ʎn ʎn + n) θ z =+ + σ + (n ʎn ʎn + n) β ʎβ ʎβ + β β ʎβ ʎβ + βθ + σ + (n ʎn ʎn + n) = θ = θ. =+ Then f MAʎ,, σ,, m, β. 21

29 Chapter two On a Certain Subclass of Multivalent Functions Conversely, suppose that f MAʎ,, σ,, m, β. We may set θ = where a is given by (2.7). then + σ + (n ʎn ʎn + n) β ʎβ ʎβ + β = z =+ fz = z a z =+ β ʎβ ʎβ + β a, + σ + (n ʎn ʎn + n) = z [z f z] =+ θ = θ z + θ f z =+ =+ = θ z + θ f z. =+ This completes the proof of theorem (2.9). θ z 22

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