On a new class of meromorphic functions associated with Mittag-Leffler function

Let X denote the class of normalized meromorphic functions f of the form

∞ f (z) = - + Hanzn                            (1.1)

n =1

defined on the unit disk

A = {z G C : 0 < |z| < 1} and which are analytic except for a set of poles of finite order on U = {z G C : |z| < 1}. Denoted by Xp and be of the form

1 M f (z) = z + ^ anZn,  an ^ 0.                          (1.2)

(c) 2025 Murugusundaramoorthy, G. and Vijaya, K.

The Hadamard product or convolution of two functions f (z) given by (1.2) and

∞

g(z) = z+^ gnzn n=1

(1.3)

is defined by

∞

(f * g)(z) := - + £ a n g n Z n .

z n=1

The function f G X with f (0) =0 is called meromorphic starlike of order й (0 C й < 1), if

^Re Z^{f >J <г S Д)}' \ J ^(z) /

(1.4)

The function f G X with f ‘ (0) =0 is called meromorphic convex of order й (0 C ^ < 1) if

" (- ^т) >” (z G ->■ f (z)

(1.5)

The class of all such functions are denoted by X k (й). Further, we denote X p (й) = X * (й) П X p and X K W = Xk (й) П X p .

Lemma 1 [1] . Suppose that й G [0,1) , r G (0,1] and the function f is of the form

∞ f (z) = z + £ bnzn, 0 < |z| < Г’

(1.6)

with b n ^ 0 . Then the condition

„ ( ^Zf '«f Re к f (z) )

> ϑ

for | z | <  r

(1.7)

is equivalent to the condition

^£ (n + й)Ь _п г ^{п +1} C 1 - й.

n =1

The condition (1.4) and the above lemma with r = 1 give the following corollary.

Corollary 1 [1] . Let f G X p be given by (1.2) . Then f G X p (й) if and only if

£ (n + й)а п C 1 - й.

n =1

(1.8)

(1.9)

Various subclasses of X have been studied rather extensively by Clunie [2], Nehari and Netanyahu [3], Pommerenke [4, 5], Royster [6], and others (cf., e.g., Bajpai [7], Mogra et al. [8], Uralegaddi and Ganigi [9], Cho et al. [10], Aouf [11], and Uralegaddi and Somanatha [12]); see also Duren [13, pp. 29 and 137], and Srivastava and Owa ([14, pp. 86 and 429], also see [11]).

Complex analysis (complex function theory) initiated in the 18th century and has since become one of the important topics in mathematics. Because of its effective applicability to a wide range of concepts and problems, this domain has significantly wedged a wide range of research areas, including engineering, physics, and mathematics. Researchers exposed some unexpected connections between ostensibly disparate study fields. Mittag-Leffler function (M-LF) research is an unusual and fascinating combination of geometry and complex analysis that deals with the structure of analytic functions in the complex domain and other domains related to sciences and engineering, has been a topic that has inspired several researchers. It was first proposed in by Mittag-Leffler [15] function ascends naturally in the solution of fractional order differential and integral equations, and exclusively in the studies of fractional generalizing of kinetic equation, random walks, L´evy flights, super-diffusive transport and in the study of complex systems. Let Eς be the function defined by

E . (z) : =

∞n v_z__

r(^n + 1), n=0

z G C, ^ G C with Re s > 0,

that was introduced by Mittag-Leffler [15] and commonly known as the Mittag-Leffler function . Wiman [16] defined a more general function E _ς,̺ generalizing the E _ς Mittag-Leffler function, that is

∞

E«(z) := E n=0

z ⁿ

—-------, z G C , s, q G C , with Re ^ > 0, Re q >  0.

Г(?п + q)

When q = 1, it is abbreviated as E . (z) = E ., i (z). Observe that the function E .^ contains many well-known functions as its special case, for example,

E i , i (z) = e ^z ,    E i , 2 (z) = e z ¹ ,    E 2 , 1 ( z ² ) = cosh z,

E 2 , i ( - z ² ) = cos z,   E 2 , 2 ( z ² ) =

sinh z

z

’    E 2 , 2 ( ^{- z} 2 ) =

E 4 (z) = 2 ( cos z ⁴ +cosh z 4 ) ,   E 3 (z) = j

ez13

+ 2e 2 ^{z 3} cos

sin z

, z

( ? ■ 3 )]

Numerous properties of Mittag-Leffler function and generalized Mittag-Leffler function can be originated, for example in [17–22]. We note that the above generalized Mittag-Leffler function Eς,̺ does not belongs to the family A , where A represents the class of functions whose members are of the form f (z) = z + V anzn, z G U, n=2

(1.10)

which are analytic in the open unit disk A and normalized by the conditions f (0) = f ‘ (0) — 1 = 0. Let S be the subclass of A whose members are univalent in A. Thus, it is expected to define the following normalization of Mittag-Leffler function as below, due to Bansal and Prajapat [18]:

∞

E .,^ ^{(z) :} z^r(Q) EU^z) = ^z + V ГмП n =2    ς

r(Q)______ n

1)+ Q) z ’

-

(1.11)

that holds for the parameters s, q G C with Re ^ > 0, Re q > 0, and z G C.

Moreover, Srivastava and Tomovski [23] function, E Tg (z~)(z G C) in the form

introduced the generalized the Mittag-Leffler

∞ ejw = E n=0

^(т ) пк     z n

r(sn + q) n! ’

(q, ^, т G C; Re(^) > max { 0, Re(K) — 1 } ; Re(K) > 0) and proved that it is an entire function in the complex z-plane, where

г(т + 0) T( (t + 1) ••• (t + 0 — 1), e = 0;

^{(т) s} = ^(7Г V,                  e = 0

is well known Pochammer symbol. Lately, Aouf and Mostafa [24] defined

T,K                 - 1 J?T,K( \       ^{Г( т + k)}    )

M, _e ^{(z)  Г}Ы z E^ ^{( z )}  Г(т)Г( _? + £)

with (q, т G C; Re(?) > max { 0, Re(K — 1 } ; Re(K > 0; Re(?) = 0 when Re(K = 1 with q = 0) and introduced a new linear operator for f G X and discussed differential inequalities for meromorphic univalent functions. Now we define a new linear operator Il’q'^m : X p ^ X p by

I^f (z)  = MT,K(z) * f (z), aT,K,m         1 ^ г(т + (n + 1)к)г(£)     n

I^   f^(z) = z ⁺ n = Г(т)Г((п + 1X + q) (n)! a n z ^, z ^G ^

where the symbol ( * ) denotes the Hadamard product (or convolution). We define a new operator IT’g’^m : X ^ X in terms of Hadamard product, as follows:

τ,κ, 0 ς, ̺

τ,κ, 1

^I ς, ̺

τ,κ,m ^I ς,̺

I^ m f (z),

(1 — 1)1^ (z) + 1(1^ f (z))’, aT,K,1 ( а^т--^ ^T-mm \

I^e    I ,Q      ( I^e  f^(z)) •

Thus,

^T,K,m ) — - + ^ (1 + (n - lW^m _ ^{Г(т + (n + 1)K)r(Q)} __ _a nn   _z          12)

I^ f^(z) = z ⁺ n =1 ^{(1 + (}n ^1)l)   Г(т)Г((п + 1X + q) (n)! a n z ^,   z ^{G       (1}-1²⁾

Shortly, we let

∞

I^^mf (z) :=- + £   a n z n ,

z n=1

where

_   ^{г(т + (}n^{+ 1)K)r(Q)}

° n = Г(т)Г((п + 1X + Q)(n)!(1 + (n 1Я ,

Г(т + 2K)r(Q)

- ¹  Г(т№ + q) •

(1.13)

(1.14)

(1.15)

One can see that 1(1’1° f (z) = zf ‘ (z) + f (z) + z ^{- 1} .

Let H be a complex Hilbert space and let L (H) denote the algebra of all bounded linear operators on H. For a complex-valued function f analytic in a domain E of the complex z-plane containing the spectrum a(P) of the bounded linear operator P, let f (P) denote the operator on H defined by [25, p. 568]

f (P) = ^ /(zl — P) ^{- 1} f(z)dz, 2ni

(1.16)

C where I is the identity operator on H and C is a positively-oriented simple rectifiable closed contour containing the spectrum a(P) in the interior domain. The operator f (P) can also be defined by the following series:

Л f⁽ⁿ⁾(o) f (P) = E f-nr^p'

n =0

which converges in the normed topology (cf. [26]).

Motivated by earlier works on meromorphic functions by function theorists (see [2, 9, 10, 12, 15, 27–30], and certain studies on Hilbert space operators [31–33] in this paper we made an attempt to define the new subclass M^ m^ , p) of E p , as given in Definition 1, related with the generalized operator I ς^τ, ̺ ^,κ,m .

Definition 1. For 0 <  $ <  1 and 0 <  p < 1, a function f G E _p be given by (1.2) is said to be in the class M^ m^ , p), if

J p (P) — 1

J r» + (1 - 2^)

< 1,

(1.17)

where

. _{m =}          P(I^^f (P)) ’

(1.18)

J )   (p - i) i_?T ^m f(P) + PP(i T^K/m f (P)) ’ ■

By fixing p = 0, we also define a new class of functions in Definition 2 and denote it by S ^T,^{K (}^^,p⁾.

Definition 2. For 0 <  ^ < 1 and 0 <  p < 1, a function f G E p be given by (1.2) is said to be in the class S^^, p), if

P ( i T^,K,m f (P)) ’ I&m f (P)

- 1

P I l'Km f (P)) ’ K'e'm f (P)

+ (1 - 2^)

< 1.

(1.19)

The present paper aims to provide a systematic investigation of the various interesting properties like coefficient inequalities, growth and distortion inequalities, as well as closure results for f in the class M^m^, p) extensively. Properties of a certain integral operator and its inverse defined on the new class M^m^, p) are also discussed.

• 2.    Coefficients Inequalities

Our first theorem gives a necessary and sufficient condition for a function f G M l^’m^ , p).

Theorem 1. Let f G E p be given by (1.2) . Then f G M T’K’m^ , p) if and only if

∞

2 {n - ^(np + p - 1) } I П а a n <  1 - ^.                   (2.1)

n =1

<1 Suppose f satisfies (2.1). Then for ||z|| = P = ri we have

H^mf (P)) ' - { (p - 1)iT ^m f (p) + p P (i <: _e ^K’^m f mnii - ^ Pf ‘ (P) + (1 - 2^) { (p - 1)1^ f (P) + pP(I™f(P)) ' }^

= 5 2 (1 - p)(n + 1)^ n a n P ^{n +1}

n =1

∞

• - - 2(1 - ^) + ^ [{1 + (1 - 2^)p } n + (1 ^- 2^)(p ^- 1)] Una n ^{p n +1} n =1

∞

C £(1 - p)(n + 1)1 5n a n h P ^{n +1} h n =1

∞

• - 2(1 - ^) + £ [{ 1 + (1 - 2^)p } n + (1 - 2

  - 1)]'3nan^Pⁿ⁺¹^ n=1

∞

= 2 £ { n - ^(np + p - 1)} ^r П а a n r ^{n +1} - 2(1 - ^) C 0, by (2.1). n =1

Hence, f satisfies (1.17), and f G M ^T,K'm (^,p). Now to prove the converse, let f G M ^T,K’m (^,p)- We need only to show that each function f of the class M l'K'm ^p) satisfies the coefficient inequality (2.1). Since f G M^'m^, p), we have by definition

PITKmf (P))'1

^’mflTH+oW TT’K’mf(z\V   1

_____ ⁽ P £££,£    J ^{( P ) +} P P^{( I} J ^{( z ))} ___________

„AfW

(p-1)iTK,mf (P+p(iTKmf (P))' + (1

< 1, z G A.

That is

_________£££-p£.+ l£aaaP£_________

- 2(1 - ^) + £“ 1 [ { 1 + (1 - 2^)p } n + (1 - 2

- 1)] Ia_na_nPⁿ⁺¹

C 1.

Since | Re(z)| C |z| = r for z G C thus by taking P = ri (0 < r < 1), from the above inequality we have

_____________En=1(1-_pKn+2X!naarn+_____________

2(1 - ^) - E“1 [{1 + (1 - 2^)p}n + (1 - 2

- 1)] Inaarrⁿ+¹

C 1,

and letting r ^ 1^-, yields the assertion (2.1) of Theorem 1. >

Fixing p = 0, we get the following.

Corollary 2. Let f G Xp be given by (1.2). Then f G ST'K(^, p) if and only if

∞

£(n + ^Kl_nan C 1 - ^.

n=1

Our next result gives the coefficient estimates for functions in M^T,K'^m(^, p).

Theorem 2. If f G MT'K'm(^, p), then

an C

1 - ^

{n - ^(np + p - 1)} In_n,

n = 1,2, 3,...

The result is sharp for the functions F_n (z) given by

^Fn^(z) = z ⁺

{n

1 - ^

- ^(np + p - 1)} ^r(n_n

_zn_,

n = 1, 2, 3,...

<1 If f G MT’'K'^m($, p), then for each n we have

∞

{n - ^(np + p - 1)} Inaa_n C £ {n - ^(np + p - 1)} an C 1 - ^. n=1

Therefore, we have

an C

Since , .    1                1 — $ n nZ   Z + {n - $(np + p - 1)HnnZ satisfies the conditions of Theorem 1, Fn(z) G Мт,к,т($, p) and the equality is attained for this function. >

For p = 0, we have the following corollary.

Corollary 3. If f G ST,K($, p), then an C

1 - $

(n + $X)\ ’

n = 1,2, 3,....

Theorem 3. If f G МТ,К^,т($, p), then

1          1 - $           ..„.„...I 1 - $ r - (1+ $ - 2$

---

The result is sharp for

1           1 - $                                          . .

^{f (z)} = z ⁺(1 + $ - 2$p)l51 z’                           ⁽²'²⁾

where 15i as given in (1.15).

<1 Since f (z) =¹+ ^n=1 anzn, we have

∞∞

Ilf (P)l C 7 + E C-+ r E a-

Taking into account the inequality

∞ an n=1

1 - $ (1+ $ - 2$p)C5i

we arrive at an estimate

"^f<^P)" ^C        .         2      ..

Similarly

If(P)l > ¹- 77—Т-$ГЕ7Г r (1 + $ - 2$p)(J.

The result is sharp for f (z) = Z + (^—^л-z. >

Similarly we have the following:

Theorem 4. If f G MT^m($, p), then

1         1 - $

Г² - (1 + $ - 2$p)75i

c»f ‘(p)icr>+

1 - $

(1+ $ - 2$p)75i ’

P = r (0 < r < 1).

The result is sharp for the function given by (2.2).

• 3.    Radius of Starlikeness

The radii of starlikeness and convexity are given by the following theorems for the class M^T^^m($,p).

Theorem 5. Let the function f belong to the class M^^m ($, p). Then f is meromorphically starlike of order p (0 C p < 1) in |z| < r.($, p,p), where ri ($’p’P) = inf n

(1 - p)[n - $(np + p - 1)]Unⁿ⁺¹

(n + p)(1 - $)

n > 1.

(3.1)

<1 Let the function f G M^l_m(p, ^) be of the form (1.2). If 0 < r < r1 (^, p, p), then by (3.1)

rn+1

_<^{(1 —}p) [^{n —}^⁽ⁿp ⁺ p-^x        (n + p)(1 - ^)

^^^^^v

1M

(3.2)

for all n G N. From (3.2) we get

∞

En + p anr

1-p n=1

n+P„n+1 <

1 — P r    <

∞

.n+1 < ^ I n=1

[ n-^(np+p-1)] On

1-й

[n — ^(np + p —

1 - ^

for all n G N, thus

—--an < 1

(3.3)

because of (2.1). If f G Xp, then by Lemma 1 the function f is meromorphically starlike of order p in |z| < r if and only if

∞

^(n + p)anrⁿ⁺¹< 1 — p.

(3.4)

n=1

Therefore, (3.3) and (3.4) give that f is meromorphically starlike of order p in |z| < r < ^r1⁽^,P,p). >

Suppose that there exists a number r, r > r1(^,p, p), such that each f G MT’K(p, ^) is meromorphically starlike of order p in |z|

1 — ^

n

(3.5)

[n — ^(np + p — 1)]33n is in the class MT’K(p,^), thus it should satisfy (3.4) with r:

∞

^(n + p)anrn+1 < 1 — p, n=1

while the left–hand side of (3.5) becomes

⁽ⁿ + p)

________^{(1 —}^)________ [n — ^(np + p — 1)]3 31

rn+1

> (n + p)

________^{(1 —}^)________ [n — ^(np + p — 1)]3 31

_x(1 — p)[n — ^(np + p — 1)]^n

— ^{1 —}p

^X (n + p)(1 — ^)

which contradicts (3.5). Therefore, the number r1(^,p,p) in Theorem 5 cannot be replaced with a grater number. This means that r1(^,p, p) is so called radius of meromorphically starlikness of order p for the class M^^p,^).

Remark 1. The above results give an improvement or better bound for order of starlikeness for f G MT^KY(^, p) compared to the results given in [32, 33].

• 4.    Neighborhoods for the Class MT’K’Y(^, p)

In this section, we determine the neighborhood for the class MT^KY(^, p), which we define as follows:

Definition 3. A function f G Xp is said to be in the class MT’K’ Y(^, p) if there exists a function g G MT K’ m(^, p) such that f (P) g(P)

<^{1 —}Y,

(z G A, 0 < y< 1)-

(4.1)

Following the earlier works on neighborhoods of analytic functions by Goodman [34] and

Ruscheweyh [35], we define the 5-neighborhood of a function f G Sp by

Ns (f) :=

g G Ep: g(z) = 1 + £bnzn and £nan - b.| < 5 .       (4.2) n=1          n=1

Theorem 6. If g G MT’K^,m(^, p) and

Y = 1 ^-

5(1 + V - 2Vp)l.h 2V(1 — p)

(4.3)

then N_s(g) С М^т^(V, p).

<1 Let f G Ns(g). Then we find from (4.2) that

∞

£ nlan — bnl < 5, n=1

which implies the coefficient inequality

∞

52 Ian — bn| < 5, n G N.

n=1

(4.4)

(4.5)

Since g G Ml^'m^, p), we have (cf. equation (2.1))

∞

£ bn < n=1

1 - V

(1 + V - 2Vp)l51 ’

(4.6)

so that         f (P)) _ _{1 <} EE1 Ian — bnl₌5(1 + V - 2Vp)U 1

g(P)      < 1 - EE1 bn        2V(1 - p)

provided y is given by (4.3). Hence, by definition, f G M^’E(V, p) for y given by (4.3), which completes the proof. >

• 5.    Closure Theorems

Let the functions Fk(z) be given by

1     СЮ

Fk (z) = z + £ fn,kzⁿ, k = 1, 2,...,m.                    (5.1)

n=1

We shall prove the following closure theorems for the class ME’EV, p).

Theorem 7. Let the function Fk(z) defined by (5.1) be in the class M^T’^K’^m(V, p) for every k = 1, 2,... , m. Then the function f (z) defined by

1     СЮ f (z) = z + ^ anzn,  an ^ ° belongs to the class Ml’K’m(V, p), whenever an = m Em=1 fn,k, n = 1, 2,...

<1 Since F_n(z) G M^m^O, p), it follows from Theorem 1 that

M

E {n - O(np + p - 1)}lПпПп,к C 1 - O n=1

for every k = 1, 2,... , m. Hence

M                    M

E {n - O(np + p - 1)} Innan = E {n - O(np + p - 1)} 1 Пп n=1                             n=1

(5.2)

(m E4

mM

= m EE^{{n - O(}np

=1 \n=1

+ p

-

1)} < n_nn_n,k^

C 1 - O.

By Theorem 1 we arrive at the required conclusion f (z) G M^mCO, p). >

Theorem 8. The class Ml^mCO, p) is closed under convex linear combination.

< Let the function Fk(z) given by (5.1) be in the class M^T’^K’m(O, p). Then it is enough to show that the function

H(z) = vF1(z) + (1 - v)F2(z), 0 C p C 1, is also in the class MT'K'm(O, p). Since for 0 C v C 1,

M

H(z) = 7 + E [vfn,1 + (1 - v)fn,2] zⁿ, n=1

we observe that

^E{n - O(np + p - 1)} IJn[vfn,1 + (1 - v)fn,2] n=1

= v^E{n - O(np + p - 1)} lfn,! + (1 - v) ^E{n - O(np + p - 1)} l^,2 C 1 - O. n=1                                    n=1

By Theorem 1, we have H(z) G MT’K’^m(O, p). >

Theorem 9. Let Fo(z) = ¹and F_n(z) f (z) G M^T’K’m(O, p) if and only if f (z) can where v_n ^ 0 and YlM=o v_n = 1.

= 1 + 7---1 ,^—nv- zn for n = 1, 2,... Then z    {n-^(np+p-1)}( n                ’ ’ be expressed in the form f (z) = M=O=o vnFn(z),

< Let

M f (z) = E vnFn(z)

n=o

Then

¹ , ^Evn^{(1 - O)}zn ^z n= ^{{n - O(}np⁺p^{- 1)}i 5}n ’

M

E n=1

________1 - O________{n - O(np + p - 1)} ^n vⁿ {n - O(np + p - 1)}^r(5_n         (1 - O)

M

= E vn = 1 - vo C 1. n=1

By Theorem 1, we have f (z) G MlKmO p).

Conversely, let f (z) G Ml’^K’m(O, p). In view of Theorem 2, we have

an C

1 - O

{n - O(np + p - 1)} ^r(n_n ’

n = 1, 2,... ,

and we may take

{n

Vn = --

- $(np + p - 1)} b.

1 - $

n an, n — 1, 2,*** ,

and Vo — 1 - ^“=1 Vn. Then f (z) — E“=₀VnFn(z). I>

• 6.    Partial Sums

Silverman [36] determined sharp lower bounds on the real part of the quotients between the normalized starlike or convex functions and their sequences of partial sums. As a natural extension, it would be interesting to look for results similar to those by Silverman for meromorphic univalent functions. In this section, motivated essentially by the work of Silverman [36] and Cho and Owa [10] we will investigate the ratio of a function of the form

∞ f (z) = z + Eanzn’(6.1)

n=1

to its sequence of partial sums

• 1                  1

• fi(z) — z and fk(z) — z + ^2 anz ,(6.2)

n=1

when the coefficients are sufficiently small to satisfy the condition analogous to

^2 {n - $(np + p - 1)} In_n an< 1 - $. n=1

For the sake of brevity we rewrite it as

∞

£ Лn|an| < 1 - $,                               (6.3)

n=1

where

Лп :—[n - $(np + p - 1)]in_n.                            (6.4)

More precisely we will determine sharp lower bounds for Re{f (z)/fk(z)} and Re{fk(z)/f (z)}. In this connection we make use of the well known results that Re |1+W(Z)} > 0 (z G A) if and only if

∞

^(z) — £ Cnzn n=1

satisfies the inequality |w(z)| ^ |z|. Unless otherwise stated, we will assume that f is of the form (1.2) and its sequence of partial sums is denoted by

k fk(z) — z + E anzn-

Theorem 10. Let f (z) G Xp($, p) given by (6.1) satisfiy condition (2.1). Then f (z) I > ^+1^,$) - 1 + $

(6.5)

Re ^ -----7----7---77----, z G U,

I ^fk^(z) J        ^Лk+1⁽P,$⁾

where

J1 — $, n = 1, 2, 3,... , k ;

(6.6)

( Лк+1(р,$), n = k + 1,k + 2,...

The result (6.5) is sharp with the function given by f (z) = 1 +

¹- $ zk+1 ^лк+1⁽р,$)

(6.7)

<1 Define the function w(z) by

1 + w(z) =Лк+1(р,$) f (z)-Лк₊1(р,$) — 1 + $

1 — w(z)      1 — $    \Jk (z)       Лк+1(р,$)

1 + £ anzⁿ⁺¹+ (       ) £ anzⁿ⁺¹

_ n=1           '          n=k+1

=                       k                          ’

1 + £ an zⁿ⁺¹

n=1

It suffices to show that |w(z)| C 1- Now, from (6.8) we can write

(6.8)

Next we estimate

w(z) =

( Ak^j ^£anzn+1 '           n=k+1

2 + 2^£anzⁿ⁺¹+ (       ) ^£anzⁿ⁺¹

n=1                      k=n+1

|w(z)|

(Лк^) £ KI

'          ' k=n+1

2 - 2 £ K| — (^K^) £ K| n=1                  n=k+1

Now ||w(P)|| C 1, if

„ / ^Лк+1(р,$) \ \\ । |       „\ I

2 I 1 — $ I / , |an| C 2 2 / , |an| n=k+1            n=1

or, equivalently,

\\ I !^Лк+1⁽Р’$⁾\\ I

/ v |an| + 1 - $           |an1 C 1’ n=1                  n=k+1

Due to the condition (2.1), it is sufficient to show that

\i , ^Лк+1(Р,$) \\ ।    ,\ ^Лп(Р>$)   ।

L |an +  1 - $   X |an CL 1 - $ |an n=1                  n=k+1       n=1

which is equivalent to

^k \ n=1

л_п(р,$) — 1 + $ 1 — $

)|anl+ \\ ( n=k+1

Лп(р_г$1—Лк±1(Ег$)

1 — $

|an|

>0.

To see that the function given by (6.7) gives the sharp result, we observe that for z = r^/k f (z) fk(z)

1 - ^

^{1 +} Лк₊1(р,^)

zⁿ^ 1 -

^{1 -}^   _ ^Лk+1⁽P^,^)^{- 1 +} ^

^лk+1⁽pp&)

^лк+х(рЛ)

, when r ^ 1 ,

which shows the bound (6.5) is the best possible for each k G N. >

7. Integral Operators

In this section, we consider integral transforms of functions in the class Mς^τ,^,̺^κ,m

(^,P)-

Theorem 11. Let the function f (z) given by (1) be in MT’Km($, p) ■ Then operator

the integral

F(z) = c J u^cf (uz) du, 0 < u< 1, 0 < c < от,

is in M^(6, p), where

8 =

(c + 2) {1 + # - 2^p} - c(1 - ^) c(1 - ^) {1 - 2p} + (1 + ^) {1 - 2p} (c + 2).

The result is sharp for the function f (z) = 1 + {i₊|L2lp}< >1z.

<1 Let f (z) G M^T’^,K’m(^, p). Then

F(z) = c J u^cf (uz) du = c J

It is sufficient to show that

X

■_nu^n+czⁿ \ du = - +

/            n=1

c

c + n + 1

fnzⁿ.

X

Ec {n n=1

^ — ⁶⁽np⁺p^-Ш?nn (c + n + 1)(1 - 6)    an ^ .

(7.1)

Since f G M^’m^d, p), we have

Ж

E n=1

{n - ^(np + p - 1)}

1 - ti

^nan ^ 1.

Note that (7.1) is satisfied, if c {n - 6(np + p - 1)} Inn < {n (c + n + 1)(1 - 6)

Rewriting the inequality, we have

^^^^^v

^(np + p - 1)} <n_n (1 - ^)

.

c {n - 6(np + p - 1)} (1 - ^) < (c + n + 1)(1 - 6) {n + ^ - ^p(1 + n)} Inn-

Solving for δ, we have

6<

(c + n +1) {n - ^(np + p - 1)} - cn(1 - ^)

= F (n).

c(1 - ^) {1 - p(1 + n)} + {(n - ^(np + p - 1)} (c + n + 1)

A simple computation will show that F(n) is increasing and F(n) ^ F(1). Using this, the results follows. >

For the choice of p = 0, we have the following result of Uralegaddi and Ganigi [9].

Corollary 4. Let the function f (z) defined by (1) be in Ep(^). Then the integral operator

F(z) = c У u^c f (uz) du, 0

0 < u< 1, 0 < c < ж,

is in Yip(5), where 6 = 1+^+7. The result is sharp for the function f (z) = 1 + z

1 -

ϑ

1 + V^Z'

Also we have the following:

Theorem 12. Let f (z), given by (1), be in MT’K’m

1                                                   1

F(z) = c [(c + 1)f (z) + zf (z)] = z + n=1

(^P),

c ⁺ n ⁺¹ fnzn,

c

c > 0.

(7.2)

Then F(z) is in MT’K’m^, p) for |z| < r(^, р,в) where r^, p, в) = inf n

-

-

(    c(1 — в) {n — ^(np + p

(1 — ^(c + n +1) {n — в(nP +

-

1)}A “

■ p^{— 1)}}       ’

n = 1, 2, 3,...

The result is sharp for the function f_n(z) =¹+

1-7

{n-7(np+p-1)}

zⁿ,    n = 1, 2, 3, . . .

w—1

w + 1 - 2в

A computation shows that this is satisfied, if

< 1.

E {n-^в(^pp+^p-lp₁+n_±l)OZ^HPHZ+I ^ 1. n=1              ^{(1 — в)c}

(7.3)

Since f G MT’K’m^, p), by Theorem 1, we have

∞

E (n n=1

The equation (7.3) is satisfied, if

— ^(np + p — 1)) In_n

1 — ^

a_n< 1-

^{{n — в(n}p ⁺p^{— 1)} (c + n +1)}       n+1   ^{n

(1 — в)c            ^z |z|    ^

— ^(np + p — 1)} ^r(П_пп_п

1 — ^

.

Solving for |z|, we get the result. >

For the choice of p = 0, we have the following result of Uralegaddi and Ganigi [9].

Corollary 5. Let the function f (z) defined by (1) be in Ep(^) and F(z) given by (7.2). Then F(z) is in Ep(^) for |z| < г^в), where

f (     ^c(1— ^{e)(n +} ^ Aⁿ⁺¹

n = 1, 2, 3,...

',     in^f V(1 — n + 1)(n + в)/    ’

The result is sharp for the function f_n(z) =¹+ Z+7 zn, ⁿ= 1, 2, 3,...

Conclusion. The interplay of geometry and analysis signifies a vital aspect of the research in the complex functions theory. The rapid progress in this area is directly related to the relationship that exists between the analytic structure and the geometric behavior of functions. In the current study, we introduced a new class of meromorphic functions that is related to the Mittag-Leffler function based on the Hilbert space operator, and we found some sufficient and necessary conditions regarding the properties of this subclass. For further research we are intended to study certain classes related to functions with respect to fixed second coefficients associated with Mittag-Leffler functions and ma jorization results.