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Let fk(z)=z+∑n=2kanzn{f}_{k}\left(z)=z+{\sum }_{n=2}^{k}{a}_{n}{z}^{n} be the sequence of partial sums of the analytic function f(z)=z+∑n=2∞anznf\left(z)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n}. In this paper, we determine sharp lower bounds for Re{f(z)/fk(z)}{\rm{Re}}\{f\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}\left(z)\}, Re{fk(z)/f(z)}{\rm{Re}}\{{f}_{k}\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}f\left(z)\}, Re{f′(z)/fk′(z)}{\rm{Re}}\{{f}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}^{^{\prime} }\left(z)\} and Re{fk′(z)/f′(z)}{\rm{Re}}\{{f}_{k}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}^{^{\prime} }\left(z)\}, where f(z)f\left(z) belongs to the subclass Jp,qm(μ,α,β){{\mathcal{J}}}_{p,q}^{m}\left(\mu ,\alpha ,\beta ) of analytic functions, defined by Sălăgean (p,q)\left(p,q)-differential operator. In addition, the inclusion relations involving Nδ(e){N}_{\delta }\left(e) of this generalized function class are considered.

In this paper, the harmonic function related to the q-Srivastava–Attiya operator is described to introduce a new class of complex harmonic functions that are orientation-preserving and univalent in the open-unit disk. We also cover some important aspects such as coefficient bounds, convolution conservation, and convexity constraints. Next, using sufficiency criteria, we calculate the sharp bounds of the real parts of the ratios of harmonic functions to their sequences of partial sums. In addition, for the first time some of the interesting implications of the q-Srivastava–Attiya operator in harmonic functions are also included. [ABSTRACT FROM AUTHOR]

In this investigation a new subclass of bi-univalent functions is established that are defined in the open unit disk A= {Z e £: |Z| < 1} and are endowed with the Sälägean type ^-difference operator. Then, Hankel inequalities for the new function class are obtained and several related consequences of the results are also stated. [ABSTRACT FROM AUTHOR]

We consider a complex-valued harmonic functions that are univalent can be written in the form f = h + g; where h and g are analytic, in a simply connected domain U and sense preserving in U, is that jh0(z)j > jg0(z)j in U: Making use of Salegean q-differential operators, we define a new subclasses harmonic starlike functions and obtain sufficient coefficient bounds, distortion theorems and extreme points for f in the new function class. Moreover, we shown that these necessary coefficient bounds are also sufficient for those functions that have negative coefficients. [ABSTRACT FROM AUTHOR]