# I. Introduction Let ?(??) be the class of analytic and pvalent function ð??"ð??"(??). The function ð??"ð??" (??) can be expressed as The Jackson's q -derivative tends to ordinary derivative when q tends to 1. The Jackson q-derivative can also be written as ?? ??,?? m ?? ?? = Î?" ?? (1+ r) Î?" ?? (1+ r?m ) ?? ???m ????????? ?? ? 0 , ?? > ?1 (1.6) A new class of multivalent function form by using Jackson Derivative Operator is defined in the following definition. Definition 5.3: A function ð??"ð??"(??) ? ?(??) is also belongs to new subclass ?? ?? ,?? ,?? (?, ?, ?, ?, q) if it follow the following condition ???? ? ???+?? ?? 2 ???? ?? ,?? ?? +?? +1 ð??"ð??"(??)? + ??(?? 2 +???? )??? ?? ,?? ?? +?? +2 ð??"ð??"(??)? (1???)??? ?? ,?? ?? +?? ð??"ð??"(??)? + ??(??+?? ?? 2 )??? ?? ,?? ?? +?? +1 ð??"ð??"(??)? ? > ?? (1.7) where ?? ? ?? 1 , ?? ? ? ? {0} , 0 ? ?? < ??, 0 ? ?? < 1, 0 ? ?? ? 1 ?????? 0 ? ?? < 1 By taking particular values of the parameters, ??, ??, ??, ??, ??, ?? we get the previously defined subclasses of univalent and multivalent function. These classes were studied by Silverman [14], Srivastava [15], Altintas et. al [2] and Khosravianarb et. al [7]. Particular Cases: 1. If ?? = 0, ?? = 0, ? = 0, ?? ? 1then from (1.7) we get ???? ? ?? ??? 1,?? ð??"ð??"(??)? + ?? ?? 2 ??? 1,?? 2 ð??"ð??"(??)? which is equivalent to ???? ? ??ð??"ð??" ? (??) + ???? 2 ð??"ð??" ?? (??) (1 ? ??)ð??"ð??"(??) + ????ð??"ð??" ? (??) ? > ?? sowe get ?? 0,?? ,?? (?, 0, ?, 0,1) ? ??(??, ??, ? , ??) and this classwas studied byAltintas et al. [2] 2. If ?? = 0, ?? = 0, ?? ? 1then from (1.7) we get ???? ? ?? ??? 1,?? ?? +1 ð??"ð??"(??)? + ?? ?? 2 ??? 1,?? ?? +2 ð??"ð??"(??)? (1 ? ??) ??? 1,?? ?? ð??"ð??"(??)? + ???? ??? 1,?? ??+1 ð??"ð??"(??)? ? > ?? so ?? 0,??,?? (?, 0, ??, ?, 1) ? ??(??, ??, ? , ??, ??) and this class was studied by Khosravianarb et al. [7] 3. If ?? = 0, ?? = 0, ? = 0, ?? ? 1, ?? = 0 then from (1.7) we get ???? ? ??ð??"ð??" ? (??) ð??"ð??"(??) ? > ?? 0 ? ?? < ?? so ?? 0,??,?? (?, 0, 0,0,1) ? T * (??, ??) and T * (??, ??) is the class of p valent starlike function of order ??. 4. If ?? = 0, ?? = 0, ? = 0, ?? ? 1, ?? = 0, ?? = 1 then from (1.7) we get ???? ? ??ð??"ð??" ? (??) ð??"ð??"(??) ? > ?? 0 ? ?? < 1 so ?? 0,?? ,1 (?, 0, 0,0,1) ? T * (1, ??) , which was earlier studied by Srivastava et al. [15]. 5. If ?? = 0, ?? = 0, ? = 0, ?? ? 1, ?? = 0, ?? = 1, ?? = 1 then from (1.7) we get ???? ? ??ð??"ð??" ? (??) ð??"ð??"(??) ? > ?? 0 ? ?? < 1 Then we get a class which was earlier discussed by Silverman [14]. 7. If ?? = 0, ?? = 0, ? = 0, ?? ? 1, ?? = 1, ?? = 1then from (1.7) we get 6. If ?? = 0, ?? = 0, ? = 0, ?? ? 1, ?? = 1then from (1.7) we get ???? ? ??ð??"ð??" ? (??) + ?? 2 ð??"ð??" ?? (??) ?? ð??"ð??" ? (??) ? > ?? 0 ? ?? < ?? which is equivalent to ???? ?1 + ??ð??"ð??" ?? (??) ð??"ð??" ? (??) ? > ?? 0 ? ?? < ?? so ?? 0,???? ?1 + ??ð??"ð??" ?? (??) ð??"ð??" ? (??) ? > ?? 0 ? ?? < 1 sowe get ?? 0,?? ,1 (?, 0, 1,0,1) ? C * (1, ??) ,which was earlier by studied Srivastava et al. [15]. 8. If ?? = 0, ?? = 0, ? = 0, ?? ? 1, ?? = 1, ?? = 1, ?? = 1 then from (1.7) we get ???? ?1 + ??ð??"ð??" ?? (??) ð??"ð??" ? (??) ? > ?? 0 ? ?? < 1 and this class of convex function was first introduced by Silverman [14]. # II. Coefficient Estimate In this part of the paper we derive the coefficient estimate of function ð??"ð??"(??), ð??"ð??"(??) ? ?? ?? ,??,?? (?, ?, ??, ?, q) Theorem 1: A function f(z) = z p ? ? a k z k ? k=n+p and f(z) ? ?(p) then f(z) belong to the class ? m ,n,p (?, ?, ?, ?, q) if and only if ? ?? ??,?? ?? ,?? ? (1+?? )[?? ?(?? +?? )] ?? ?1????? +?? [?? ?(?? +?? +1)] ?? ???? (1???) (1+?? )[???(?? +?? )] ?? ?1????? +?? [???(?? +??+1)] ?? ???? (1???) ? ?? ?? ? 1 ? ?? =??+?? (2.1) where ?? ??,?? ?? ,?? = Î?" ?? (1+?? )Î?" ?? (1+???(?? +?? )) Î?" ?? (1+??)Î?" ?? (1+???(?? +?? )) ?? ? ?? 1 , ?? ? ? ? {0} , 0 ? ?? < ??, 0 ? ?? < 1, 0 ? ?? ? 1 ?????? 0 ? ?? < 1 Proof: Let us consider that ð??"ð??"(??) ? ?? ?? ,?? ,?? (?, ?, ??, ?, q) so we have ???? ? ???+?? ?? 2 ???? ?? ,?? ?? +?? +1 ð??"ð??"(??)? + ?? (?? 2 +???? )??? ?? ,?? ?? +?? +2 ð??"ð??"(??)? (1???)??? ?? ,?? ?? +?? ð??"ð??"(??)? + ?? (??+?? ?? 2 )??? ?? ,?? ?? +?? +1 ð??"ð??"(??)? ? > ?? Since ð??"ð??"(??) = ?? ?? ? ? ?? ?? ?? ?? ? ?? =?? +?? and ?? ??,?? ?? +?? ð??"ð??"(??) = Î?" ?? (1+??) Î?" ?? (1+???(?? +?? )) ?? ???(?? +??) ? ? Î?" ?? (1+??) Î?" ?? (1+???(?? +?? )) ?? ?? ? ?? =?? +?? ?? ???(?? +??) (2.2) so we have ?? ??,?? ?? +?? +1 = Î?" ?? (1+??) Î?" ?? (???(?? +?? )) ?? ???(?? +?? +1) ? ? Î?" ?? (1+??) Î?" ?? (???(?? +?? )) ?? ?? ? ??=?? +?? ?? ???(?? +?? +1) (2.3) ?? ??,?? ?? +??+2 = Î?" ?? (1+??) Î?" ?? (???(?? +?? +1)) ?? ???(?? +?? +2) ? ? Î?" ?? (1+??) Î?" ?? (???(?? +?? +1)) ?? ?? ? ?? =?? +?? ?? ???(?? +??+2) (2.4) By using (2.2), (2.3) and (2.4) in (2.1) then we get numerator and denominator of (2.1) as numerator is denoted by N and denominator by D N = (?? + ???? 2 ) ? Î?" ?? (1+??) Î?" ?? ????(?? +?? )? ?? ???(?? +??+1) ? ? Î?" ?? (1+??) Î?" ?? ??? ?(?? +?? )? ?? ?? ? ?? =??+?? ?? ???(?? +?? +1) ? + ??(?? 2 + ????) ? Î?" ?? (1 + ??) Î?" ?? (?? ? (?? + ?? + 1)) ?? ???(?? +?? +2) ? ? Î?" ?? (1 + ??) Î?" ?? (?? ? (?? + ?? + 1)) ?? ?? ? ??=?? +?? ?? ?? ?(?? +??+2) ? D = (1 ? ??) ? Î?" ?? (1+??) Î?" ?? (1+???(?? +?? )) ?? ???(?? +??) ? ? Î?" ?? (1+??) Î?" ?? (1+???(?? +?? )) ?? ?? ? ?? =?? +?? ?? ?? ?(?? +?? ) ? + ?? (?? + ???? 2 ) ? Î?" ?? (1 + ??) Î?" ?? ??? ? (?? + ??)? ?? ???(?? +??+1) ? ? Î?" ?? (1 + ??) Î?" ?? ??? ? (?? + ??)? ?? ?? ? ?? =?? +?? ?? ???(?? +?? +1) ? solve above by using [??] ?? = Î?" ?? (1+?? ) Î?" ?? (?? ) and on considering the value of z to be real and let ?? ? 1 then we get Î?" ?? (1 + ??) Î?" ?? ?1 + ?? ? (?? + ??)? ?(1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??)? ? ? Î?" ?? (1 + ??) Î?" ?? ?1 + ?? ? (?? + ??)? ?? ?? ?(1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??=??+?? ? ??(1 ? ??)? on simplifying we get, ? ?? ??,?? ?? ,?? ? (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) ? ?? ?? ? 1 ? ?? =??+?? where ?? ??,?? ?? ,?? = Î?" ?? (1+?? )Î?" ?? (1+???(?? +?? )) Î?" ?? (1+??)Î?" ?? (1+???(?? +?? )) Conversely: Let us assume the inequality (2.1) is true To Prove: ð??"ð??"(??) ? ?? ?? ,??,?? (?, ?, ??, ?, q), for this we have to show that ? > ?? which implies ð??"ð??"(??) ? ?? ?? ,?? ,?? (?, ?, ??, ?, q) so, the proof of theorem 1 is completed Corollary 1: Let the function ð??"ð??"(??) = ?? ?? ? ? ?? ?? ?? ?? ? ?? =??+?? is a member of new subclass ?? ?? ,?? ,?? (?, ?, ??, ?, q) of multivalent function then ?? ?? ? ? (1+?? )[???(?? +?? )] ?? ?1? ???? + ??[???(?? +??+1)] ?? ???? (1???) (1+?? )[???(?? +?? )] ?? ?1? ???? + ??[???(?? +?? +1)] ?? ???? (1???) ? 1 ?? ?? ,?? ?? ,?? (2.9) where k = n + p, p is some natural number, n is a natural number. # III. Property of New Subclass Related to Radii of Star Likeness, Convexity and Close to Convexity In this part of the paper, we derive some results related to Radii of starlikeness, convexity and close to convexity for the function ð??"ð??"(??) belonging to the new subclass ?? ?? ,?? ,?? (?, ?, ??, ?, q) ð??"ð??"(??) ? ?? ?? ,?? ,?? (?, ?, ??, ?, q) and ð??"ð??"(?? ) = ?? ?? ? ? ?? ?? ?? ?? ? ?? =??+?? To prove is close to convex of order ?? ; 0 ? ?? < ?? in |??| < ?? 1 * for this we have to show that ? ð??"ð??" ? (??) ?? ?? ?1 ? ??? ? ?? ? ?? |??| < ?? 1 * (3.2) ? ð??"ð??" ? (??) ?? ???1 ? ??? = ? ???? ???1 ? ? ???? ?? ? ?? =?? + ?? ?? ???1 ?? ???1 ? ??? = ? ? ???? ?? ? ??=?? +?? ?? ???1 ?? ???1 ? ? ? ???? ?? ? ??=?? +?? |??| ?? ??? (3.3) The inequality (3.2) is less than or equal to ?? ? ?? if ? ? ?? ????? ? ?? ?? ? ?? =?? + ?? |??| ????? ? 1 (3.4) we know that ð??"ð??"(??) ? ?? ?? ,??,?? (?, ?, ??, ?, q) if and only if ? ?? ??,?? ?? ,?? ? (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) ? ?? ?? ? 1 ? ?? =?? +?? The inequality (3.2) is hold true if ? ?? ?? ? ?? ? |??| ????? ? ?? ?? ,?? ?? ,?? ? (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) ? or, we have Let ð??"ð??"(??) |??| ????? ? ? ????? ?? ? ?? ??,?? ?? ,?? ? (1+?? )[?? ?(?? +?? )] ?? ?1????? + ??[?? ?(?? +?? +1)] ?? ????(1???)(Notes |??| < ?? 1 * = ????ð??"ð??" ?? ? ?? + ?? ?? ?? ? ?? ?? ? ? (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ??(1 ? ??)? (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ??(1 ? ??)? ? ?? ??,?? ?? ,?? ? 1 ????? Hence, the given function ð??"ð??"(??) is p-valent close to convex of order ?? Theorem 3: Let the function ð??"ð??"(??) = ?? ?? ? ? ?? ?? ?? ?? ? ?? =?? +?? and ð??"ð??"(??) ? ?? ?? ,?? ,?? (?, ?, ??, ?, q) then the function ð??"ð??"(??) is a p-valent starlike of order ?? ; 0 ? ?? < ?? in |??| < ?? 2 * ,where ?? 2 * = ????ð??"ð??" ?? ? ?? + ?? ?? ????? ?? ??? ? ? (1+?? )[???(?? +?? )] ?? ?1????+?[?? ?(?? +?? +1)] ?? ????(1??) (1+?? )[???(?? +?? )] ?? ?1????+?[???(?? +?? +1)] ?? ???? (1??) ? ?? ??,?? ?? ,?? ? 1 ?? ??? (3.6) Proof: ð??"ð??"(??) ? ?? ?? ,?? ,?? (?, ?, ??, ?, q) and ð??"ð??"(??) = ?? ?? ? ? ?? ?? ?? ?? ? ?? =?? +?? To prove the function ð??"ð??"(??) is p-valent starlike of order ??; 0 ? ?? < ?? in |??| < ?? 2 * for this we have to show that ? ??ð??"ð??" ? (??) ð??"ð??"(??) ? ??? ? ?? ? ?? |??| < ?? 2 *(3.7) Now we take the L.H.S. part of the inequality (3.7) ? ??ð??"ð??" ? (??) ð??"ð??"(??) ? ??? = ? ??(???? ???1 ? ? ???? ?? ? ?? =?? +?? ?? ???1 ) ?? ?? ? ? ?? ?? ?? ?? ? ?? =?? +?? ? ??? = ? ? (?? ? ??)?? ?? ?? ?? ? ?? = ??+?? ?? ?? ? ? ?? ?? ?? ?? ? ?? =?? + ?? ? ? ? (?????)?? ?? |??| ?? ??? ? ?? =?? +?? 1?? ?? ?? |??| ?? ??? ? ?? = ?? +??(3.8) The inequality (3.7) is less than or equal to ?? ? ?? if ? (?????) (?????) ?? ?? ? ?? = ?? + ?? |??| ????? ? 1 (3.9) we know that ð??"ð??"(??) ? ?? ?? ,??,?? (?, ?, ??, ?, q) if and only if ? ?? ??,? ??ð??"ð??" ?? (??) ð??"ð??" ? (??) + (1 ? ??)? = ? ??(??(?? ? 1)?? ???2 ? ? ??(?? ? 1)?? ?? ? ?? =?? +?? ?? ???2 ) ???? ???1 ? ? ???? ?? ?? ?? ?1 ? ?? = ?? +?? + (1 ? ??)? = ? ? ??(?? ? ??)?? ?? ? ?? = ?? +?? ?? ????? ?? ? ? ???? ?? ?? ?? ??? ? ?? =?? +??? ? ??(?? ? ??)?? ?? |??| ????? ? ?? =?? +?? ?? ? ? ???? ?? |??| ?? ??? ? ?? =?? +?? The inequality (3.12) is less than or equal to ?? ? ?? if ? ?? (????) ??(???? ) ?? ?? ? ?? =?? +?? |??| ????? ? 1 (3.13) we know that ð??"ð??"(??) ? ?? ?? ,??,?? (?, ?, ??, ?, q) if and only if ? ?? ??,?? ?? ,?? ? (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) ? ?? ?? ? 1 ? ?? =?? +?? The inequality (3.12) is hold true if ![some natural number, ?? ? ? The function ð??"ð??"(??) defined in (1.1) isan analytic function and pvalent function in the open unit disc?? 1 = {?? ? |??| < 1} If a function ð??"ð??"(??) ? ?(??) satisfies the following condition ???? ? ??ð??"ð??" ? (??) ð??"ð??"(??) ? > ?? ?? ? ?? 1 , 0 ? ? < ?? , ?? ? ? (1.2)then ð??"ð??"(??) is a p -valent starlike function of order ? and if a function ð??"ð??"(??) ? ?(??) satisfies the following condition???? ?1 + ??ð??"ð??" ?? (??) ð??"ð??" ? (??) ? > ? ?? ? ?? 1 , 0 ? ? < ?? , ?? ? ? (1.3) then ð??"ð??"(??) is a p -valent convexfunction of order ?To define a new subclass of multivalent function by using Jackson derivative, we use the following definitions Definition 1:Letð??"ð??"(??) = ?? ?? ? ? ?? ?? ?? ?? ? ?? =?? +?? ?????? ð??"ð??"(??) = ?? ?? ? ? ?? ?? ?? ?? ? ?? =??+??are the members of the class ?(??), then their convolution product or Hadamard product is defined as(ð??"ð??" * ð??"ð??")(??) = (ð??"ð??" * ð??"ð??")(??) = ?? ?? ? ? ?? ?? ?? ?? ?? ?? ? ?? =?? +?? (1.4) and generally the convolution product of functions ð??"ð??"(??) and ð??"ð??"(??) is denoted by (ð??"ð??" * ð??"ð??")(??) or(ð??"ð??" * ð??"ð??")(??). Definition 2: The Jackson q-derivative of a function ð??"ð??"(??) is denoted by ?? ?? ð??"ð??"(??) ???? ?? ??,?? ð??"ð??"(??) and it is defined as ?? ??,?? ð??"ð??"(??) = ð??"ð??"(??)?ð??"ð??"(???? ) ??????? ?? ? 0 ?????? ?? ? 1 (1.5)](image-2.png "") 2![Let the function ð??"ð??"(??) = ?? ?? ? ? ?? ?? ?? ?? ? ?? =?? +?? and ð??"ð??"(??) belong to ?? ?? ,?? ,?? (?, ?, ??, ?, q) then the function ð??"ð??"(??) is p-valent close to convex of order ?? ; 0 ? ?? < ?? in |??| < ?? 1 * , where [?? ?(?? +?? )] ?? ?1? ???? + ??[?? ?(?? +?? +1)] ?? ????(1???) (1+?? )[???(?? +?? )] ?? ?1 ? ???? + ??[???(?? +?? +1)] ?? ????(1??? )](image-3.png "Theorem 2 :") ![1+?? )[???(?? +?? )] ?? ?1????? + ??[???(?? +?? +1)] ?? ???? (1??? ) ? (3.5) so we get the required result A New Subclass of Multivalent Function Defined by using Jackson Derivative Operator](image-4.png "") ![?? ?? ,?? ? (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??)(1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) ? ?? ?? ? 1 ? ?? =?? +?? The inequality (3.9) is hold true if ? |??| ????? ? ?? ?? ,?? ?? ,?? ? (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) ?or, we have|??| ????? ? ? ????? ????? ? ?? ??,?? ?? ,?? ? (1+?? )[???(?? +?? )] ?? ?1????? + ??[???(?? +?? +1)] ?? ???? (1???) (1+?? )[???(?? +?? )] ?? ?1????? + ??[???(?? +??+1)] ?? ???? (1???) ? (3.10)so we get the required result|??| < ?? 2 * = ????ð??"ð??" ?? ? ?? + ?? ?? ?? ? ?? ?? ? ?? ? ? (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) ? ?? ??,?? ?? ,?? ? 1 ?????Hence, the given function ð??"ð??"(??) is p-valent starlike of order ?? Theorem 4: Let the function ð??"ð??"(??) = ?? ?? ? ? ?? ?? ?? ?? ? ?? =?? + ?? and ð??"ð??"(??) ? ?? ?? ,?? ,?? (?, ?, ??, ?, q) then the given function ð??"ð??"(??) is a p-valent convex function of order ?? ; 0 ? ?? < ?? in |??| < ?? 3 [?? ?(?? +??)] ?? ?1????? + ??[?? ?(?? +??+1)] ?? ???? (1???) (1+?? )[???(?? +??)] ?? ?1????? + ??[???(?? +?? +1)] ?? ????(1???) ? ?? ??,?? ?? ,?? ? 1 ?? ??? (3.11) ð??"ð??"(??) ? ?? ?? ,??,?? (?, ?, ??, ?, q) and ð??"ð??"(??) = ?? ?? ? ? ?? ?? ?? ?? ? ?? =?? +?? To prove the function ð??"ð??"(??) is p-valent convex function of order ?? ; 0 ? ?? < ?? in |??| < ?? 3 * for this we have to show that ? ??ð??"ð??" ?? (??) ð??"ð??" ? (??) + (1 ? ??)? ? ?? ? ?? |??| < ?? 3 * (3.12) Taking the L.H.S. part of the inequality (3.12)](image-5.png "") ![New Subclass of Multivalent Function Defined by using Jackson Derivative Operator](image-6.png "?A") ![](image-7.png "Notes") ![? |??| ????? ? ?? ?? ,?? ?? ,?? ? (1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ?? [?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??)(1 + ??)[?? ? (?? + ??)] ?? ?1 ? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 [???(?? +?? )] ?? ?1????? + ??[???(?? +?? +1)] ?? ???? (1???) (1+?? )[???(?? +?? )] ?? ?1????? + ??[???(?? +??+1)] ?? ???? (1???) ??)[?? ?(?? + ??)] ?? ?1? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) (1 + ??)[?? ?(?? + ??)] ?? ?1? ???? + ??[?? ? (?? + ?? + 1)] ?? ? ? ??(1 ? ??) ??? ??,?? ?? ,?? ? 1 ?????Hence, the given function ð??"ð??"(??) is p-valent convex function of order ?? References Références Referencias A New Subclass of Multivalent Function Defined by using Jackson Derivative Operator , O., ?zkan, ?., & Srivastava, H. M. (2000), Neighborhoods of a class of analytic functions with negative coefficients, Applied Mathematics Letters, 13(3), 63-67. doi:10.1016/S08939659(99)00187-1. 2. Altinta?, O., Irmak, H., & Srivastava, H. M. (1995), Fractional calculus and certain starlike functions with negative coefficients, Computers & Mathematics with Applications, 30(2), pp 9-15. doi: 10.1016/0898-1221(95)00073-8 3. Aouf, M.K. & Mostafa, A.O. (2008), On a subclass of n-p-valent prestarlike functions, Comput. Math. Appl., 55, pp 851-861. doi.org/10.1016/j.camwa.2007. 05.010](image-8.png "") (2.5)Let L = |?? ? (1 + ??)|Refand?? =(1???)??? ?? ,?? ?? +?? ð??"ð??"(??)? + ??(??+?? ?? 2 )??? ?? ,?? ?? +?? +1 ð??"ð??"(??)?(2.6)4. Aqlan, E.S. (2004), Some Problems Connected with Geometric Function Theory,Ph.D. Thesis, Pune University, Pune.L= ? and K = |?? + (1 ? ??)| ???+?? ?? 2 ???? ?? ,?? ?? +?? +1 ð??"ð??"(??)? + ?? (?? 2 +???? )??? ?? ,?? ?? +?? +2 ð??"ð??"(??)? (1???)??? ?? ,?? ?? +?? ð??"ð??" (??)? + ??(??+?? ?? 2 )??? ?? ,?? ?? +?? +1 ð??"ð??"(??)? K = ?? (1 + ??)?(2.7)1 Year 2022 5 Global Journal of Science Frontier Research Volume XXII Issue ersion I V V ( F )© 2022 Global Journals???? ? (?? + ???? 2 ) ??? ??,?? ?? +?? +1 ð??"ð??"(??)? + ?? (?? 2 + ????) ??? ??,?? ?? +?? +2 ð??"ð??"(??)? (1 ? ??) ??? ??,?? ?? +?? ð??"ð??"(??)? + ??(?? + ???? 2 ) ??? ??,?? ?? +?? +1 ð??"ð??"(??)? ? > ?? According to Lemma [4] A if ?? = ?? + ???? then ???? ?? ? ?? â??" |?? ? (1 + ??)| ? |?? + (1 ? ??)| ???+?? ?? 2 ???? ?? ,?? ?? +?? +1 ð??"ð??"(??)? ?? (?? 2 +???? )??? ?? ,?? ?? +?? +2 ð??"ð??"(??)? (1???)??? ?? ,?? ?? +?? ð??"ð??"(??)? + ?? (??+?? ?? 2 )??? ?? ,?? ?? +?? +1 ð??"ð??"(??)? + (1 ? ??)? (2.8) From (2.7) and (2.8), ?? ? ?? > 0 i.e.|?? + (1 ? ??)| ? |?? ? (1 + ??)| > 0 which implies ????(??) > ?? Hence the inequality ???? ? ???+?? ?? 2 ???? ?? ,?? ?? +?? +1 ð??"ð??"(??)? + ?? (?? 2 +???? )??? ?? ,?? ?? +?? +2 ð??"ð??"(??)? 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