Bad hack to do this,but

1. Open \server\odoo\addons\mail\static\src\xml\systray.xml

2. Comment out the following:

<!--li class="o_mail_navbar_item">            
    <a href="#" title="Inbox">                
    <i class="fa fa-at"/> <span class="o_notification_counter"/>            
    </a>
</li-->

and

<!--li class="o_mail_navbar_item">
 <a class="dropdown-toggle" data-toggle="dropdown" aria-expanded="false" title="Conversations" href="#">
     <i class="fa fa-comments"/> <span class="o_notification_counter"/>            
 </a>
 <ul class="o_mail_navbar_dropdown dropdown-menu" role="menu">                
    <li class="o_mail_navbar_dropdown_top">                    
        <div>                        
            <button type="button" class="btn btn-sm o_filter_button o_selected"> All </button>                        
            <button type="button" class="btn btn-sm o_filter_button" data-filter='chat'> Chat </button>                        
            <button type="button" class="btn btn-sm o_filter_button" data-filter='channels'> Channels </button>                    
        </div>                    
    <button type="button" class="btn btn-sm o_new_message"> New message </button>                
    </li>                
    <li class="o_mail_navbar_dropdown_channels"/>            
 </ul>        
</li-->