A study on a class of $p$-valent functions associated with generalized hypergeometric functions

Let A p be the class of functions f (z) normalized by

∞ f (z) = zp +    ap+nzp+n, p E N,(1.1)

n=1

winch are analytic and p-valent in tlie unit disc U. Let f and g be analytic in the open unit disc U. We say that f is majorized by g iii U and write f (z) « g(z) (z E U),(1.2)

if there exists a function y. anah-tic In U such that

Kz)| 6 1,   f(z) = +(z)g(z) (z E U)-(1-3)

It may be noted here that (1.2) is closely related to the concept of quasi-subordination between analytic functions.

For f (z) aiid g(z) are ana lytic in U, we say that f is subordinate to g if there exists the Schwarz function ш, analytic in U, with ш(0) = 0 aiid |w(z)| < 1 such that f (z) = g(w(z)). z E U. We denote tins subordination by f (z) + g(z). If g(z) is miiv;dent in U, then the subordination is equivalent to f (0) = g(0) aiid f (U) C g(U).

If f (z) aiid g(z) beloiig to Ap, then the Hadamard product f * g is defined by f (z) * g(z) = zp + XX ap+nbp+nz^+П p E N.

n=1

El-Ashwah [2] studied the following p-valent function, which defined by generalized hypergeometric functions rG.(abbx; zp) = zp + XX (“j“ "' a)n zp+n, p € N,

П= ^(b 1 ) n ••• ^(b s)n ^n!

where a i E C, bq E C\{0, — 1, -2,...}, (i = 1,..., r, q = 1,..., s), and r 6 s + 1; r, s E No, and (x)_n is the Pochhammer symbol defined by

_ r^x+n) _ j

(X)n      Г(х)      )

• 1,                            n = 0,

x(x + 1) • • • (x + n — 1), n = {1, 2, 3,...}.

Let Lm^2 ,p E A p is defined by

∞

Vm,b     p , X

^L X i ,x 2 ,p z ⁺ z^

n =1

p + (Ax + A2)n + b p + A2n + b

z p + n

p E N,

where m, b E N_o = N U {0}. A2 > A_x > 0.

Corresponding to _rG_s(a1 ,b_x; zp), Lm’b 2 , _p and using the Hadamard product, we define a new generalized differential operator DmX^p^A) as follows:

Definition 1.1. Let f E Ap, then a generalized differential operator Dm^p^Af (z) : A p ^ Ap is given as

DmA2,p(ai,bi)f (z) = (rGs(ax,bi; zp) * L ^Xx2» * f (z))

∞

= zp + X n=1

(1.4)

(1.5)

P + (Ax + A2)n + b 1^m (ax)n ... (ar)n ap+nzP+П p + A2n + b    J (bx )n ... (bs)n n!

It follows from the above definition that

(p + A2n + b) D^^alA)f (z)

= (p + A2n — pAx + b) Dm^,p(aX,b1)/ (z) + ^(DmX^alA)f (z))0.

Remark 1.1. It should be remarked that the linear operator D ^X 2(ax,bx)f(z) ^ » generalization of many operators considered earlier. Let us see some of the examples:

For A2 = b = 0, the operator Dm’X2 p(ax,bx)f reduces to the operator was given by Selvaraj and Karthikeyan [1].’ ’

For m = 0, the operator Dm’X2 p(ax,bx)f reduces to the operator was given by El-

Ashwah [2].’ ’

For m = 0 and p = 1, the operator Dm’X2 p(ax,bx)f reduces to the well-known operator introduced by Dziok and Srivastava [3].’ ’

For A2 = b = 0 an dp = 1, we get the operator studied by Selvaraj and Karthikeyan [4].

For m = 0, r = 2, s = 1 an dp = 1, we obtain the operator which was given by Hohlov [5].

For r = 1. s = 0. ax = 1. Ax = 1. A2 = b = 0 aiid p = 1, we get the Salageaii derivative operator [6].

For r = 1. s = 0. ax = 1. A2 = b = 0 aiid p = 1, we get the generalized Salageaii derivative operator introduced by Al-Oboudi [7].

For m = 0. r = 1. s = 0. ax = d + 1 aiid p = 1, we obtain the operator introduced by Ruscheweyh [8].

For r = 1, s = 0, ax = d + 1 an dp = 1, we obtain the operator studied by El-Yagubi and Darus [9].

For m = 0. r = 2 aiid s = 1. a2 = 1 aiid p = 1, we obtain the operator studied by Carlson and Shaffer [10].

For r = 1, s = 0, a1 = 1, A2 = 0 and p = 1, we get the operator introduced by Catas [11].

Next, by using the generalized differential operator Dm^,p(a1 , b1), we study the class Smbjjai, bi, A, B,7] as follows:

Definition 1.2. Let f e A_p, then f e SmAjJan b1 ,A,B,7] оf p-valent functions of complex order 7 = 0 in U, if and only if

1 1+1 z^^^bf-j^ p + Л1   . A z ∈ U,

(1.6)

I y (Dmb3>i,bi)fMj   p       1+Bz where p e N. m,b,j e No = N U {0}. 7 e C\{0}. A2 > A1 > 0. —1 6 B < A 6 1, ai e C. bq e C\{0, —1, —2,...} (i = 1,..., r. q = 1,..., s) aiid r 6 s + 1: r, s e No.

Clearly, we have the following relationships:

• (i)    when m = 0. p = 1. j = 0. r = 2. s = 1. ai = b1. a₂ = 1. A = 1 aiid B = —1, then the class Smb^H ,b₁,A,B,7] reduces to the class S (7).

• (ii)    when m = 0. p = 1. j = 1. r = 2. s = 1. ai = b1. a₂ = 1. A = 1 aiid B = —1, then the class Sm’^b ,b₁,A,B,7] reduces to the class C (7).

• (iii)    when m = 0. p = 1. j = 0. r = 2. s = 1. a1 = b1. a₂ = 1. A = 1. B = —1 and 7 = 1 — a, then the class Sm’bj _p[a1,b1, A,B,7] reduces to the class S*(a) for 0 < a < 1.

• 2.    Majorization Problem

The classes S ( y ) and C (7) are said to be classes of starlike and convex of complex order 7 = 0 111 U, were considered by Nasr and Aonf [12] and S*(a) denote the class of starlike αU

A majorization problem for functions f belong to the class S^’b^j p[ai,bi, A, B,7] is considered.                                                   ’’ ’

Theorem. Let f e Ap and sup])ose that g e Smm^ p[a1,b1 ,A,B,7]•

(Dmbx^xAa^bi^{)f (z))(j) is ma}j°^{rized ь}у ⁽D’mi^^tbx₂’P^(ai^,biMzTj i ¹¹U ^then

I(Dm1+x;bp(ai,bi)f(z^^\6\(Dmi+x2bp(ai,bi)g(z))(j)|   for lzl 6 ro,(2.1)

where ro = ro(p, 7, Ai, A2, b, A, B ) is the smallest positive root of the equation

3 / »         (p + А2П + b\ „

-

p ⁺ A ² n ^{+ b} +2^|]_r2 A1

r3 7(A — B) + ----2 B

A1

-

(A         ( p + А2П + b\ „ , J , ( p + А2П + b\ _n

(2.2)

(2.3)

7(A — B) — ----г----- B +2 r + ----- = 0,

V Ai / J \     Ai/

—1 6 B < A 6 1;  A₂ > A1 > 0;   b e N_o;   P e N; 7 e C\{0}.

<1 Since g e S m^j _p[ai, bi, A, B,7 ] we can get from (1.6), that

1 + 1 / ^z(Dm₁’^bA2’P⁽ai^,bi "Wy)^¹ — + .! = 1 + Aw(z)

• ⁷     (Dm_i’bA2’p(ai,bi )g(z))^(j)      p /     ^{1 +} Bw^(z),

where 7 € C\{0}- j,P € N P > j and w D ^anai^'tie in U with

w(0) = 0,   |w(z)| < 1   (z € U).

From (2.3), we get hDmh>A)^

(DmU^b i Mz-Y'

By noting that

(P — j) + [Y (A — B ) + ( p — j )B]w(z)

1 + Bw(z)

.

(2.4)

m,b               (j+i)    pP ^{+ A} 2 ^{n + b}

z^yaiM(z)   =1^--Ai--

+ P - j -

p + X₂n + b ^A i

) re^ nj _n ,p ⁽a i ^,b i ^)f (z)^ ,

(2.5)

and by virtue of (2.4) and (2.5) we get

Iwh ,,(« i ,bMzY)^m |

6 ____________ p +Y+ b [1 + |B|z|]

—I (Dm+inpgcz))⁰ ’ |.       i²-⁶)

|B| |z|

p + X n n+b Ai

Next, since (D mm^b _{X n},_p(a_i,b_i)f (z))^(j) is majorized by U.) m\X _n,_p(a_i, b_i)g(z))^(j) in the unit disc U, thus from (1.3) we have

(Dmh„(«bW' li 'j¹ = . D ■ 2 ,p (a₁.^b1)9(z)У^,j■

Differentiating it with respect to z and multiplying by z we get z(Dmh „pf^Lb i )f (z))°^+ti

Now by using (2.5) in the above equation, it yields

ИрА^Ь^ )f (zV)⁾⁽ j

+ b(z)(D m 1 +i ₂ ’ ^b p (ai,b i )g(z))^(j) .

(2.7)

^zb(^z)(^DmX .ppObbiNzaj p + X n n + b Ai

Thus, by noting that ^ € D satisfies the inequality

Ib⁰(z)l 6 ¹ - ^2^|2 (z € U), ¹ - ^|z|

(2.8)

by using (2.6) and (2.8) in (2.7), we get

| (Dm^ha-b i )f (z))j |

|b(z)| +

1 -|b(z)|2

|z|(1 + |B||z|)

1 -|z|2

(p^n^) -|Y(A - B) +

( p+Xnn+b i Ai

)|B|||z|

I (D m1+t;‘P (al.b 1 )g(^z))^t ’⁾ | ,

(2.9)

which upon setting

|z| = r arid |^(z)| = p (0 6 p 6 1)

leads us to the inequality

|(D m+d (ai,bi f I²»® |

(1 — r2)

p + X ^ n + b λ 1

ФМ___________

Y (A — B) + ( ^{p + X} 2 n ^{+ b} \ B

r

D - I

(2.10)

where

ф(p) = -r(1 + |B|r)p² + (1 - r²)

x

p + A2n + b ^Ar

—

Y(A - B) + f p ⁺ ^²n ⁺ Vr P + r(1 + |B|r) V     ^Ar     ) J

(2.П)

takes its maximum value at p = 1 with rg   = rg(p, 7, Ar, A2,b, A, B). Here rr(p,Y,Ar ,A2,b,A,B) is the smallest positive root of the equation (2.2).

Furthermore, if 0 6 p 6 r_r(p, 7, A_r, A₂, b, A, B). then the function ^(p) defined by

ФМ = -a^{(1 +} |B mp^{2 + (1 —} ct ²⁾|

(2.12)

/p + A2n + bA . ..          (p + A2n + bA .        _       .

X -----;------ - | y ( A — B) + -----;------ B | ct p + ^(1 + | B | ct )

Ar                               Ar is seen to be an increasing function on the interval 0 6 p 6 1, so that

ФЫ 6 ^⁽¹⁾ = ^{(1 —} ct ^{2 )}

p + A2 n + b Ar

—

p+A n+b

y ( A — B) + ----7----- B ct

Ar

(2.13)

0 6 p 6 1 (0 6 ct 6 rr(p,Y,Ar, A2,b, A, B)).

Hence upon setting p = 1 in (2.10) we conclude that (2.1) of Theorem 2.1 holds true for |z| 6 rr(p,Y,Ar,A2,A,B) where rr(p,Y,Ar,A2,A,B) is the smallest positive root of equation (2.2).

Putting A = 1 and B = —1 in Theorem 2.1, we have the following result:

Corollary 2.1. Let f G A p and supiэозе that g G S m’b ^j _p(a_r, b_r,Y). If (D m 1 , _p(^or,br)f (z))^{( j )} is majorized by (D J 1 , _p(ar,br)g(z))^{( j )} in U, then

where

rg = rg(p,Y, Ar, A2, b) =

k — ^у — 4( ^n+ ) I 2 y — ( Е+Лз^

2|27 — (

p + X ₂ n + b Ai

p+A n+b         p+A n+b k = 2 + ----7----- + 2y — ----7-----

Ar                        Ar p G N; Y,Ar G C\{0}, b G Ng, A2 > 0

and S m’b( ^j _E(a_r, b_r,Y) be a special case of S m’b( ^j _p[a_r, b_r, A, B, 7] when A = 1 an d B = — 1.

Setting p = 1. m = A2 = b = 0. A_i = 1. j = 0. r = 2. s = 1. ai = b_i aird a₂ = 1 in Corollary 2.1, we get the following corollary:

Corollary 2.2. Let f E Ap and sup} rose that g E S ( y )• ^ f (z) ^JS ana jo vised lay g(z) in U. then

If0(z)| 6 lg'(z)l (|z| <гз)- where

3+ |2y - 1| - P9 + 2|2y - 1| + |2y - 1|2 r0 = r0 W =2i2g-i which is a. known result obtained by Altintas et al. [13].

For y = 1, the Corollary 2.2 reduces to the following result:

Corollary 2.3. Let f (zj E Ap and sup}Dose that g E S* = S*(0). If f (zj is majorized by g(z) in U. thou

|f0(z)|6 |g0(z)| (|z|6 2 -V3), which is a known result obtained by MacGregor [14].

• 3.    Coefficient Estimates

The coefficient estimate for the class Sm^j _р[а 1 -Ь 1 -A-B-Y] is obtained, when j = 0.

Definition 3.1. Let Sm^ ,Да₁,b₁ -A-B-y ] denote the subclass of p-valent functions which satisfy the condition

1 +                                   pl ^         - y( DmU<4 >W (z)    ) 1 + Bz where p E N. y E C\{0}. A2 > Ai > 0. m-b E No- —1 6 B < A 6 1. ai E C. bq E

C\{0- —1- —2-...}- (i = 1-... - r- q = 1-... - s)- and r 6 s + 1; r-s E N o .

Theorem 3.1. Let f E Ap. If it satisfies the condition:

Pro г                           1 h p+(Ai +A2)n+b]m (ai)n^^^(ar )n n=i n + |Y(A B) nB|      p+A2n+b       (bi)n-(bs)n n! |aP+n|

6 1-

(3.2)

I y I( a - в )

then f E S mm^b_x2 , _p[ai-bi-A-B-Y]•

C Let f E S m^b _{X 2} , _p[a_i-b_i-A-B-Y]- then we can write (3.1) as follows:

1+1 z(D m ^b _A2 , p (a i -b i )f (z))⁰ Y     D mbA ₂ , p (a i -b i )f (z)

-

p

1 + Aw(z)

1 + Bw(z)

which gives zCDmk.pbgbJfW Dmby2, p(ai-bi )f (z)

-p = Y(A - B) - B

z(D m_xl, p (a i -b i )f (z))⁰ D m ^b A ₂ , p (a i -b i )f (z)

-

P^j w(z).

(3.3)

From (3.3), we obtain

p V Гp+(Ai +A2)n + b 1m (ai)n^^^(ar)n                  p+n pz +         p+Aan+b       (bi)n•••(bs)n n! (p + n)ap+nz n=1

~P    P ГP+(A1 + A2)n + b 1m (aih^Xar)n „    p+nn z +         p+Aan+b       (bi)n—(bs)n n! ap+nz n=1

-p

∞

Y(A - B) - B

∞

P P p+(A1 + A2)n+b pz + ^    p+Aan+b n = 1 L                 J

n=1

m Jo i lnjjjlorjn                 p+n

(b i ) n ••• (b s ) n n! ^{(p + n)}a p ⁺ n ^z

~P    V ГP+(Ai +A2)n + b 1m (ai■ •••'a-)n „     „p+n z +          p+Aan+b        (bi)n•••(bs)n n! ap+nz n=1

-

p

> w(z)

which yields

P    Г p+(A i +A 2 )n+b 1 m (a i )n^(a r ) n        n

• 2-    n    p+A a n+b        ( b i )n-(b s ) n n! a p + n z

n=1

∞

P p+(A i +A a )n+b 1

1 +        p+Aan+b n=1

^(a i ) n •"^(a r ) n           n

(b i ) n ••• (b s ) n n! a p+n z

Y ^{( a - B})

-

B

V Г p+(Ai+A2 )n+b 1m (ai )n •••(ar )n         n n          p+A2n+b        (bi )n ...(bs)n n! ap+nz n=1

^V Г p+iA i +A ab +b 1 ^{m (} a i ) n ••• (a r ) n         n

1+         p+Aan+b       (bi)n-(bs)n n! ap+nz n=1

> w(z).

Since |w(z)| 6 1,

∞ X n=1

∞

n

p + (A 1 + A 2 )n + b

p + A 2 n + b

m (a 1 ) n • • • (a r ) n           _n

(b 1 ) n ••• (b s ) n n! a^p+nz

6 y(a - B) - y^vBn n=1

- y ( A - в )]

p + (A i + A 2 )n + b

p + А 2 П + b

m ^(a 1 ) n • • • ^(a r ) n           _n

(b 1 ) n ••• (b s ) n n! a^p+nz

.

Letting |z| ^ 1 through real values, we have

∞

X ^{[n +} |Y⁽A ^{- B) -} Bn^|]

n=1

p + ( A 1 + A 2 ) n + b ^m

p + А 2 П + b

^(a 1 ) n • • • ^(a r ) n

(6 1 ) . ••• (b s ) n n!^|ap+ " ^{16 |7|(A}    B )■

therefore,

EVJn + I y ( a - B) - Bn|] [ p+pX+b]’'*

⁽ a i ) n ^^^⁽ a r ) n

(b i ) n ••• (b s ) n n! ^|ap+n

.

I y I( a - в )

Remark 3.1. Other works related to different classes of p-valent functions can be found in [15, 16].