1 Basic inequality For any real numbers ρ d satisfying ρ 1 = 1, and for any natural number m, we have d|m µ(d) d|m ρ d 2 = e|m λ e , λ e = [d 1 ,d 2 ]=e ρ d 1 ρ d 2. Our goal is to optimize the choice of (ρ d) d1. Let m = (n, P (z)), multiply by a n and sum over n using (g): say. Minimizing XG + R is quite difficult. However, if we restrict the support of ρ to d √ D, it is relatively easy to mi...

In this paper, we first determine Bohr’s inequality for the class of harmonic mappings \(f=h+\overline{g}\) in unit disk \(\mathbb {D}\), where either both \(h(z)=\sum _{n=0}^{\infty }a_{pn+m}z^{pn+m}\) and \(g(z)=\sum }b_{pn+m}z^{pn+m}\) are analytic bounded or satisfies condition \(|g'(z)|\le d|h'(z)|\) {D}\backslash \{0\}\) some \(d\in [0,1]\) h is bounded. particular, obtain p-symmetric map...

Figure S1. Cont. 68 PnLac1 95 PoLac 95 PsLac 95 PpLac 95 LsLac 65 HmLac . M M M M . . V V V V . . L L L L . . S S S S . . T T T T . . K K K K . . L L L L . . A A V V . . A A A A . . L L L L . . V V V V . . A A A A . . S S S S . . L L L L . . P P P P . . F F F F . . V V V V . . L L F F . . A A A A . . A A V V . . T T T T . . K K K K . . K K K K . . L L L L . . D D D D . . F F F F . . H H H H . ....

1896 H a s c h e k a n d L i n d e n t h a l (32) o p a c i f i e d v e s s e l s o f a n a m p u t a t e d h a n d u s i n g T e i c h m a n n ' s m i x t u r e . B r a u s (11) d e m o n s t r a t e d v e i n s a n d a r t e r i e s o f a h a n d u s i n g m e r c u r y . 1907 K a s s a b i a n (40) d e m o n s t r a t e d v e s s e l u t i l i z i n g b i s m u t h s u b n i t r a t e a n d ...

(a) Let d = (m,n). Then notice that d · nd = n ≡ 0 ∈ Z/n, so nd is in the kernel of the map ·m. Now let k ∈ Z/n satisfy mk = 0 ∈ Z/n. Then we have mk = c′n for some c′ ∈ Z. We can write n = d · nd , and so we have mk = cdd . Dividing both sides by d, we find m d k = c ′ n d where m d ∈ Z. But md is relatively prime to n, and so letting k = ( m d )−1 c′ nd , we see that k ∈ {n/d} and we are done...